Kirchberg and Rordam asked whether three notions of pure
infiniteness agree for Cstar-algebras that are not simple: ordinary
pure infiniteness, meaning every positive element is properly
infinite in stable Cuntz comparison; strong pure infiniteness,
meaning two positives and one arbitrary element can be pushed into
nearly orthogonal corners at the same time; and absorption of the
Cuntz algebra O_inf. Two of the three reductions are settled below,
for arbitrary algebras in the ordinary-to-strong direction and for
exact algebras in the fixed-amplification direction, with a second
independent argument in the nuclear case.
Notation is standard and used without further comment. C, R, N
are the complex, real and natural numbers; M_n the n-by-n complex
matrices; C(X) the continuous complex functions on a compact space;
B(H) the bounded operators on a Hilbert space; a* the adjoint; A_+
the positive elements (a = b*b for some b); K the compact operators
on a separable infinite-dimensional Hilbert space; A** the
enveloping von Neumann algebra; e_h the support projection of h, in
A**; h^(+n) the direct sum of n copies of h; @min the spatial
(minimal) tensor product; Prim(A) the primitive ideal space and a_P
the image of a in A/P; O_inf the Cuntz algebra on countably many
isometries with pairwise orthogonal ranges. All algebras are
complex, may be nonunital, and may be nonseparable; a representation
of a unital algebra is unit-preserving, hence nondegenerate.
You already know three kinds of objects that behave alike:
(a) the complex numbers C;
(b) the n-by-n complex matrices M_n;
(c) the continuous complex functions C(X) on a compact space X,
for instance X = [0,1].
In each you can add, multiply, multiply by scalars, take an
adjoint (conjugate, conjugate transpose, complex conjugate), and
measure size with a norm. Each has a unit: 1, the identity
matrix, the constant function 1. A Cstar-algebra is what these
three have in common.
A Cstar-algebra is a set A with the operations +, ., scalar
multiplication by C, an adjoint a -> a*, a norm ||.||, and one
distinguished element 1, such that
(i) A is a complex vector space and 1 is a unit for . ;
(ii) multiplication is associative and complex-bilinear;
(iii) (a*)* = a, (ab)* = b* a*, (a + lb)* = a* + conj(l) b*;
(iv) ||ab|| <= ||a|| ||b|| and ||1|| = 1;
(v) ||a* a|| = ||a||^2 (the Cstar identity);
(vi) A is complete for the norm.
Axiom (v) is the rigid one. Together with completeness it forces
every Cstar-algebra to be, up to isomorphism, a norm-closed
*-subalgebra of some B(H). That theorem is never used; the axioms
suffice, and every construction below is an explicit
algebraic formula.
An element a is self-adjoint if a* = a. It is positive if a = b*b
for some b; equivalently a is self-adjoint with spectrum in
[0,infinity). Write a >= 0, A_+ for the positives, and a <= b
for b - a >= 0. Every positive has a positive square root a^(1/2),
and every self-adjoint a splits as a = a_+ - a_- with a_+, a_- >= 0
orthogonal; a_+ = (a)_+ is the positive part.
If A has no unit, the unitization A~ = A + C 1 is the smallest
unital algebra containing A as an ideal. Most theorems below allow
nonunital and nonseparable algebras.
An IDEAL is a linear subspace J closed under multiplication by A
from either side and closed in the norm; it is proper if J is not
all of A. Ideals behave as in the three examples: in C(X) they are
the functions vanishing on a closed subset; in M_n there are only
0 and M_n; in C there are only 0 and C.
For h >= 0 let
Her(h) = closure of hAh = closure of h^(1/2) A h^(1/2) ,
Id(h) = the closed two-sided ideal generated by h.
Her(h) is a hereditary subalgebra: any positive bounded above by a
positive of Her(h) also lies in Her(h). Two facts are used
constantly.
FACTS.
(a) The closure of A h^(1/2) A equals Id(h). In words, the ideal
generated by h is exactly the span of things you can sandwich
through h^(1/2). This is continuity of multiplication.
(b) u in Her(h) if and only if u is A-supported entirely inside
the "support corner" supp(h) A** supp(h). The support
projection supp(h) is taken in the double dual A**, which
contains A.
The mental picture: h^(1/2) is a map into a "subspace", Her(h) is
the matryoshka of operators living on that subspace, and Id(h) is
the ideal of all algebra operations that pass through it.
A representation of A on a Hilbert space H is a *-homomorphism
pi: A -> B(H). It is irreducible if the only closed subspaces of
H invariant under every pi(a) are 0 and H. A one-dimensional
irreducible representation is the same thing as a CHARACTER: a
nonzero *-homomorphism A -> C. For instance M_n has no characters
once n >= 2 (proof in the sibling account); C(X) has one character
per point, chi(f) = f(x0).
Irreducible representations are the atoms of the theory. The set
of them, modulo unitary equivalence, is the SPECTRUM. Its quotient
by "same kernel" is the PRIMITIVE IDEAL SPACE Prim(A): the points
are the kernels P of irreducible representations, equivalently the
primitive ideals. For commutative A = C(X) it is just X. For a
general nonsimple algebra Prim(A) can be a tangled non-Hausdorff
space, but it controls everything: an element is detected (is
nonzero) somewhere if and only if it survives some irreducible
representation.
For P in Prim(A) and a in A write a_P for the image in the
primitive quotient A/P. Because A/P embeds faithfully in the
operators of an irreducible representation, ||a_P|| is the operator
norm of a in that irreducible representation. Two slices will
matter:
Omega(c,alpha) = { P : ||c_P|| > alpha } (open),
K(c,alpha) = { P : ||c_P|| >= alpha } (quasi-compact).
The word "quasi-compact" (compact without asking Hausdorff) means
every open cover has a finite subcover; K(c,alpha) has it even when
Prim(A) does not (Lemma 1.9). This is the reason
the exact route never needs a separable or nice primitive space.
Fix a positive h. In M_n(N) with n huge (so there is room) you can
place h twice side by side: h (+), h, the block diagonal
diag(h, h). A finite-rank projection cannot do this "inside
itself", but projections of infinite rank can: a countably infinite
projection is unitarily equivalent to two copies of itself. That
splitting is exactly what "infiniteness" should mean in operator
algebra, made stable so that no size bookkeeping gets in the way.
Working inside A, allow a little slack and access to the compact
operators K. Recall that A is always regarded with K attached:
every h sits in A in the top-left corner of A (x) K, and matrix
sizes are absorbed by K.
DEFINITION (properly infinite). h in A_+ is PROPERLY INFINITE if
its two-fold direct sum is Cuntz below it:
h (+), h precsim h,
i.e. the block diagonal diag(h,h) sits, in Cuntz comparison, below
the single element h. We
count 0 as properly infinite, for bookkeeping.
Think of it as "h is at least twice itself". For projections this
says exactly "infinite rank"; for a general positive it says the
support of every cutoff can be split in two inside itself. The
whole paper begins with the observation that whether a positive
element can contain two copies of itself, once asked uniformly, is
a much stronger and much more useful hypothesis than it looks.
This is the nonsimple, nonunital version of pure infiniteness due
to Kirchberg and Rordam: for a Cstar-algebra, "every positive
element properly infinite" is EQUIVALENT to being purely infinite
in the usual nonsimple sense. That equivalence is not needed as a
hypothesis; consequences are proved directly.
Let A be a Cstar-algebra and h, g in A_+, with matrix sizes allowed
and absorbed by K.
DEFINITION (Cuntz comparison). Write h precsim g, "h is Cuntz
below g", if for every eps > 0 there is an x (of matching
rectangular matrix size over A) with
|| h - x* g x || <= eps.
Write h ~_Cu g, "Cuntz equivalent", if h precsim g and
g precsim h. The class of h is [h].
So h precsim g means h can be squeezed out of g by a rectangular
transform, up to arbitrarily small norm error. It is a semigroup
order: [h] <= [g] if h precsim g, and addition is direct sum,
[h] + [g] = [h (+), g]. Four elementary rules are used throughout.
They are proved as Lemmas 1.1 and 1.2 and quoted here so
that the tour is honest about what is being used.
RULE (rectangular absorption). [x*x] = [x x*] for every
rectangular matrix x over A. ("a rectangular corner and its
opposite corner have the same Cuntz class".)
RULE (orthogonal sum). If c_1, ..., c_r >= 0 are pairwise
ORTHOGONAL (c_i c_j = 0 for i not j), then
[c_1 + ... + c_r] = [c_1] + ... + [c_r].
RULE (stability under maps and tensor factors). Cuntz comparison
survives every *-homomorphism, and survives tensoring with a fixed
positive c, with the error multiplied by ||c||.
RULE (top class). [h] <= [s] whenever s is positive and h is in
Her(s); in particular [h] <= [1_A] for h in A_+ with A unital.
A note on order. h precsim g is a statement about h and g, not
about any ambient product; it is invariant under passing between A
and A (x) K. This is why it is legitimate to compare mismatched
matrix sizes: pad with zeros inside K.
Two regularity words, both standard, both only ever used as we
state them.
TENSOR PRODUCTS. For Cstar-algebras A, B the algebraic tensor
product A odot B carries several Cstar norms. The two extremes
are:
the MAXIMAL norm ||.||_max, the largest completion, where the
two factors interact as much as any representations allow;
the MINIMAL (spatial) norm ||.||_min, the smallest completion,
where each factor sees only its own Hilbert space; it is the
completion A @min B.
Always ||.||_min <= ||.||_max.
NUCLEAR. A is NUCLEAR if the two norms agree for every B, i.e.
A @min B = A (x) B for all B (no ambiguity of norm). Matrix
algebras, commutative algebras, and quotients and tensor products
of nuclear algebras are nuclear. For a nuclear algebra the
tensor product symbol needs no subscript.
EXACT. Some sequences of algebras 0 -> J -> E -> E/J -> 0 stay
"isometric" after tensoring. Precisely, A is EXACT if for every
closed two-sided ideal I of every Cstar-algebra C, the canonical
map
(A @min C) / (A @min I) -> A @min (C/I)
is an ISOMORPHISM (not only injective). A nuclear algebra is
exact, because both sides collapse to the maximal tensor product.
Exactness passes to subalgebras, to matrix amplifications M_n(A),
and (a deeper fact, not needed on the direct route) to quotients.
The point of exactness here is quantitative. Non-exactness of a
test algebra will supply a FIXED tiny error threshold
below which a certain "reducing copy" construction is impossible
in any unital exact algebra. Nuclearity gives an alternative,
independent nuclear route.
Let A be a complex Cstar-algebra. Two conditions about positive
elements are the whole subject.
CONDITION (P) -- "individual proper infiniteness".
EVERY h in A_+ is properly infinite:
h (+), h precsim h for all h in A_+.
CONDITION (F_n) -- "weak pure infiniteness, fixed amplification".
There is ONE integer n >= 1, chosen for the whole algebra,
such that for EVERY h in A_+ the n-fold direct sum is properly
infinite:
(h^(+n)) (+), (h^(+n)) precsim h^(+n) for all h,
for this single fixed n.
This is written (F_n). The integer n is fixed BEFORE any positive
element is examined; it therefore applies to every scalar spectral
cutoff used later. (P) is exactly (F_n) with n = 1.
The third, strongest, condition is not about single positives but
about a positive and an arbitrary ELEMENT at once. It is the
standard definition of strong pure infiniteness.
CONDITION (STRONG PURE INFINITENESS, KS form).
For all x, y in A_+ and every eps > 0 there exist s, t in A
with
|| s* x^2 s - x^2 || < eps,
|| t* y^2 t - y^2 || < eps,
|| s* x y t || < eps.
The first two lines say "x and y can each be approximately
recovered through their own corners"; the third says those two
corners can be made ALMOST MUTUALLY ANNIHILATING at the same time.
No norm bound on s, t is part of the definition.
History in one breath. (P) => (F_n) is trivial (n = 1). Strong
pure infiniteness implies (P). Whether (P), (F_n) for some n, and
strong pure infiniteness all agree is the comparison question
Kirchberg-Rordam asked; they agree in the simple, real-rank-zero,
Hausdorff-primitive, dimension-zero, and Z-stable cases known to
them. The reduction below holds for arbitrary algebras on the
ordinary-to-strong side, and for the whole question on exact
(hence nuclear) algebras on the amplification side.
O_inf is the Cstar-algebra generated by isometries s_1, s_2, ...
with pairwise orthogonal ranges, s_i* s_j = delta_ij and the
ranges summing to 1. Up to isomorphism it does not depend on the
countable list. It is simple, nuclear, purely infinite, separable,
and it is self-absorbing: O_inf @min O_inf ~ O_inf.
DEFINITION (absorption). A is O_inf-ABSORBING if
A ~_* A @min O_inf.
O_inf-absorption is one of the cleanest regularity properties a
separable nuclear algebra can have; Kirchberg-Rordam proved it is
equivalent to strong pure infiniteness for separable nuclear
algebras that are stable or have approximate units of projections,
and Kirchberg's central-sequence results extend this to all
separable nuclear algebras, unital or not. Nothing below uses more
than those stated equivalences, and no further hypothesis is added.
Every word is now defined.
THEOREM A (individual => strong; arbitrary algebras).
Let A be a complex Cstar-algebra satisfying (P). Then A is
strongly purely infinite. In fact, for all a, b in A_+, all
c in A, and every eps > 0, there are s, t in A with
|| s* a s - a || < eps, || t* b t - b || < eps,
|| s* c t || < eps.
Taking (a, b, c) = (x^2, y^2, xy) recovers strong pure
infiniteness. The cross term c is ARBITRARY, not a product of the
diagonal inputs; that is the strength of the statement. Moreover
the s, t one gets for positive contractions have norms bounded
independently of how small eps is.
THEOREM B (exact amplification removal).
Let A be a complex Cstar-algebra that is exact and satisfies
(F_n) for one fixed n >= 1. Then every positive element of A is
properly infinite (A satisfies (P)); consequently, by Theorem A,
A is strongly purely infinite.
COROLLARY (absorption). Let A be a separable nuclear complex
Cstar-algebra satisfying (F_n) for one fixed n. Then A is
O_inf-absorbing: A ~_* A @min O_inf.
For separable nuclear algebras this says WEAK pure infiniteness
(F_n for some fixed n), pure infiniteness (P), strong pure
infiniteness, and O_inf-absorption are all equivalent. The single
finite amplification test recognises a tensorial regularity. The
reverse arrow absorption => strong pure infinity is the classical
Kirchberg-Rordam direction; the new content is the forward arrow.
Two constants deserve a warning. In (F_n) the integer n is fixed
once and for all for the algebra under discussion. It appears in the
number of graph entries, in the amplification M_n(A) the uniform gap
is applied to, and in the tuple the nuclear route transports. It is
never chosen after seeing a positive element.
Two further constants enter. gamma is the universal error threshold
from the non-exactness of the universal contraction algebra, and sigma
is a fixed constant below min(1, gamma/3). Both are fixed at the
start of the uniform-gap argument and reused unchanged later, and the point of the uniform perturbation is that sigma
depends on none of them: not on n, not on the algebra, not on the
slice.
The shared toolbox comes first, and everything in it is proved. The
two load-bearing tools are the comparison rules (Lemma 1.2) and the
transport lemma (Lemma 1.3); the rest supports them.
Every comparison below is STABLE: finite positive matrices over A
are placed in the common stabilization A (x) K by padding with
zeros in the extra coordinates, and "positive" allows matrix sizes
throughout. For h >= 0 in A (or a matrix amplification) put
Her(h) = closure of hAh = A intersected with e_h A** e_h,
Id(h) = the closed two-sided ideal generated by h,
e_h = 1_(0,inf)(h) in A**, the SUPPORT projection of h,
and write supp(h) = e_h. Here A** is the enveloping von Neumann
algebra (the double dual); it contains A, it is strong-operator
closed, and it is where support projections and polar
decompositions of rectangular carriers naturally live. Whenever a
constructed thing is asserted to lie in A (not only in A**), that
is said.
Support bookkeeping. For h >= 0:
e_h h = h = h e_h; e_h f(h) = f(h) for continuous f on
[0,inf) with f(0) = 0; e_h = 1 (the identity) if h is
invertible in A~.
The universal (normal) extension of a representation pi: A -> B(H)
to A** -> B(H) sends e_h to e_{pi(h)}; this follows from normality
applied to h(h + 1/j)^(-1) increasing to e_h. Consequently, in a
representation, "support" means the same thing whether computed
before or after applying pi. When a unit or a support projection is
used inside a DEGENERATE representation, one restricts to its
essential subspace; this does not change the norm of any represented
algebra element.
Lemma 1.2 (elementary comparison rules). Let D be a Cstar-algebra.
(i) For h in D_+: Her_D(h) = D intersected with e_h D** e_h.
If 0 <= u <= K v for u, v in D_+ and a constant K >= 0, then
u is in Her(v) and u precsim v. More generally EVERY positive
element of Her(v) is Cuntz subequivalent to v.
(ii) For a rectangular matrix z over D one has z*z ~_Cu zz*.
Cuntz subequivalence is transitive, is preserved by block
sums, and is closed under norm limits in its FIRST variable
when the second is fixed. Every *-homomorphism preserves
Cuntz sub- and equivalence after amplification. For finite
positives d, v, the stable comparison d precsim v is
EQUIVALENT to: there exist r_j (matching rectangular matrices
over D) with r_j* v r_j -> d in norm.
(iii) IDEAL COMPARISON. If d, v are positive in a common
stabilization, d lies in Id(v), and v is properly infinite,
then d precsim v.
Proof of (i). First the bidual description. The inclusion "Her(h)
contained in the support corner" holds on hDh and passes to closure.
Conversely let u in D_+ with e_h u = u. Choose continuous positive
contractions f_j, f_j(0) = 0, increasing pointwise to 1 on (0,inf).
On the compact state space of the unitization, the continuous
functions phi -> phi(u (1 - f_j(h)) u) decrease pointwise to 0
(normality evaluates the strong limit f_j(h) -> e_h). Dini's
theorem upgrades this to uniform convergence. The positive norm
formula plus (1 - f_j(h))^2 <= 1 - f_j(h) then give
|| (1 - f_j(h)) u || -> 0, so
u = lim_j f_j(h) u f_j(h) in Her(h).
(The corner statement f_j(h) in Her(h) follows by cutting f_j off
near 0.) A general element supported on both sides by e_h reduces
to the positive case on the positive and negative parts of its real
and imaginary parts.
Next, 0 <= u <= K v. Compressing by 1 - e_v gives
u^(1/2)(1 - e_v) = 0, so u is supported under e_v, i.e. u in
Her(v). If u in Her(v)_+, take a cutoff approximate identity g_j(v)
of Her(v), g_j vanishing near 0. The quotient below is continuous
when extended by 0 at 0, and
x_j = g_j(v) v^(-1/2) u^(1/2) in D,
x_j* v x_j = u^(1/2) g_j(v)^2 u^(1/2) -> u.
Hence u precsim v. This covers the "more generally" clause, since
Her(v)_+ is exactly {u in D_+ : u <= K v for some K}. []
Proof of (ii). Rectangular. With z over D put
x_j = z (z*z + 1/j)^(-1/2). Then x_j*(zz*)x_j = (z*z)^2 (z*z +
1/j)^(-1), which tends to z*z in norm, so z*z precsim zz*. The same
with z* gives zz* precsim z*z, so they are Cuntz equivalent. The
inverses live in the unitization and are multiplied by algebra
elements, keeping the x_j in D (or its matrix amplification).
Transitivity: if u precsim v via x and v precsim w via y, then
(yx)* w (yx) = x*(y*wy)x is close to x*vx is close to u, so u
precsim w after rescaling x by max(1, ||x||^2). Block sums: block
diagonal coefficients. Norm limits in the first variable: a
diagonal choice of coefficients across an approximating sequence.
Homomorphisms: if phi: D -> E and x_j* v x_j -> d, then
(phi(x_j))*(phi(v))(phi(x_j)) = phi(x_j* v x_j) -> phi(d), so phi
preserves subequivalence; applying in both directions preserves
equivalence. Finite rectangularity: let P_n, P_m be scalar
coordinate projections in the multiplier algebra of D (x) K. If
x_j* v x_j -> d in the stabilization, then r_j = P_m x_j P_n lie in
the rectangular matrices over D and r_j* v r_j = P_n x_j* v x_j P_n
-> d. Conversely rectangular coefficients embed in the
stabilization. Multiplier compressions keep algebra membership
with no unit; the argument covers zero matrices. []
Proof of (iii). Work in the common stabilization, call it D. The
closed span of D v^(1/2) D is Id(v), so d^(1/2) is a norm limit of
finite sums sum_{i=1}^r x_i v^(1/2) y_i. For such a sum set
X = [x_1 ... x_r] in M_(1,r)(D), Y = [y_1; ...; y_r] in
M_(r,1)(D), V = diag(v, ..., v) in M_r(D), z = X V^(1/2) Y.
The order and rectangular rules give
z*z <= ||X||^2 (Y* V Y),
Y* V Y ~_Cu V^(1/2) (Y Y*) V^(1/2) <= ||Y||^2 V,
hence z*z precsim V. Proper infiniteness of v plus block addition
and induction give V precsim v for every finite r. Now z*z -> d
along the approximating sums, and the norm-limit rule gives
d precsim v. The proof covers v = 0. []
Consequences used silently later. Cuntz equivalence preserves
proper infiniteness, and a block sum of finitely many properly
infinite positives is properly infinite (block addition plus
transitivity). Also u precsim v gives Id(u) contained in Id(v),
because every comparison approximant lies in Id(v).
This is the engine of every "put several orthogonal copies inside
one hereditary algebra" step. A Cuntz comparison, after a spectral
cutoff, is implemented by ONE rectangular carrier w (living in the
bidual) whose products with the cutoff hereditary algebra stay in
the ORIGINAL algebra and whose conjugation preserves norms in EVERY
representation, also at matrix levels.
Lemma 1.3 (transport after a cutoff). Let A be a Cstar-algebra,
m, n >= 1, d in M_n(A)_+, v in M_m(A)_+. Suppose d precsim v in
A (x) K. For eps > 0 put h = (d - eps)_+. Then there is
w in M_(m,n)(A**) with
(a) w*w = e_h and w w* <= e_v;
(b) w Her_{M_n(A)}(h) is contained in M_(m,n)(A),
w Her_{M_n(A)}(h) w* is contained in Her_{M_m(A)}(v);
(c) conjugation by w is an injective *-homomorphism on Her(h);
(d) for l in Her(h)_+: (w l w* - tau)_+ = w (l - tau)_+ w*
and w (l - tau)_+ w* ~_Cu (l - tau)_+ for every tau >= 0;
(e) w Her(l) w* = Her(w l w*);
(f) every representation pi, with matching matrix amplifications,
satisfies || pi^(m)(w k w*) || = || pi^(n)(k) || for
k in Her(h).
The products in (b)-(f) are algebra elements; w ITSELF is only
asserted to lie in the bidual.
Proof. Compress a stabilization comparison coefficient to the
first m target and first n source coordinates. These scalar
multiplier projections give s in M_(m,n)(A) with
|| d - s* v s || < eps/2. Put z = v^(1/2) s h. Order in the
unitization and functional calculus for d (using h d h >= eps h^2
on the support of h) give
(eps/2) h^2 <= z*z = h s* v s h <= (||d|| + eps/2) h^2.
By Lemma 1.2(i) the two inequalities force Her(z*z) = Her(h);
consequently Her(|z|) = Her(h) and e_(z*z) = e_h. Let z = w |z| be
the rectangular polar decomposition in the bidual. Then
w*w = e_h. The factorization z = v^(1/2) (s h) gives
w w* = e_(zz*) <= e_v. This is (a).
For k in Her(h) = Her(|z|): take cutoff functions g_j for |z|, zero
near 0, with g_j(|z|) k -> k. Since
w g_j(|z|) = z g_j(|z|) / |z| lies in M_(m,n)(A),
norm closure gives w k in M_(m,n)(A). If k >= 0 then
w k w* = (w k^(1/2))(w k^(1/2))* is in M_m(A); it is supported under
e_v (by (a)), hence in Her(v). Linear decomposition handles general
k. This is (b). The identity w*w = e_h gives (c): multiplicativity
and injectivity. It also gives (d): functional calculus vanishing at
0 commutes with the homomorphism, and the two squares of
w (l - tau)_+^(1/2) are the asserted positives, with Cuntz
equivalence by Lemma 1.2(ii).
For (e): normal functional calculus gives e_(w l w*) = w e_l w*.
So the image of Her(l) is supported under e_(w l w*), which gives one
inclusion by Lemma 1.2(i). Conversely a sandwich by the image of l
has the form
(w l w*) b (w l w*) = w ( (w l)* b (w l) ) w*,
and the inner algebra element is supported on BOTH sides by e_l,
hence lies in Her(l); conjugation is an injective *-homomorphism on
Her(l), so the image is norm closed, and taking closures gives the
reverse inclusion.
Finally (f): the normal extension of pi carries (a) to the same
identities with represented supports; conjugating back by the
represented w* recovers pi^(n)(k); the two contractive conjugations
give both inequalities. []
Split a carrier for a direct sum. If d_0 in M_l(A)_+ and
d = d_0^(+r) in M_(rl)(A)_+, split w in M_(m,rl)(A**) into block
columns w_1, ..., w_r in M_(m,l)(A**). Then
w_i* w_j = delta_ij e_((d_0 - eps)_+).
The r copies have orthogonal ranges and the SAME initial support; the
single rectangular map transports every entry of a matrix over the
common hereditary algebra, at every amplification. Concretely, for
every j >= 1 the support identities apply to 1_j (x) w in
M_(jm,jn)(A**), so for K in M_j(Her_{M_n(A)}(h)) entrywise
conjugation gives an algebra matrix and preserves its norm in every
representation (Lemma 1.3(f)), with matching jm, jn amplifications.
Sums versus block sums. For orthogonal c_1, ..., c_r in A_+ the row
R = [c_1^(1/2) ... c_r^(1/2)] in M_(1,r)(A) has
R*R = diag(c_1, ..., c_r) and R R* = sum_i c_i, so by Lemma 1.2(ii)
[c_1 + ... + c_r] = [c_1 (+), ... (+), c_r] = [c_1] + ... + [c_r].
Moreover f(sum_i c_i) = sum_i f(c_i) for continuous f with f(0) = 0
(first for polynomials with zero constant term, then by functional
calculus). Applying this to the orthogonal transported copies
theta_i(y) = w_i y w_i* of y in Her_(M_l(A))((d_0 - eps)_+)_+, the
cutoff and comparison clauses of Lemma 1.3 give, for every tau >= 0,
( sum_{i=1}^r theta_i(y) - tau )_+
~_Cu ( (y - tau)_+ )^(+r).
This includes tau = 0. []
Lemma 1.5 (ideals and spectral cutoffs). Let D be a Cstar-algebra.
(i) If J = closure( union over lambda of J_lambda ) for an
increasing directed family of closed ideals, and z in J_+,
then for every delta > 0 the cutoff (z - delta)_+ lies in
some J_lambda. In particular the ideals Id((z - delta)_+),
directed as delta decreases to 0, densely exhaust Id(z).
(ii) PRIMITIVE IDEAL TEST. For u, v in D_+, u lies in Id(v) if
and only if every irreducible representation pi with
pi(v) = 0 also has pi(u) = 0.
(iii) If f, g are continuous nonnegative functions of a positive a,
vanish at 0, and have the same open positivity set in the
spectrum of a, then Her(f(a)) = Her(g(a)) and f(a) ~_Cu g(a).
Proof. (i) Choose b in J_lambda with ||z - b|| < delta; then
||z + J_lambda|| < delta, and functional calculus in the quotient
gives (z - delta)_+ in J_lambda. As (z - delta)_+ -> z, the stated
exhaustion follows. (ii) Apply separation of points by irreducible
representations to the quotient D/Id(v). (iii) The two support
projections coincide (same open positivity set, both vanish at 0);
Lemma 1.2(i) gives equal hereditary algebras and mutual comparison,
hence Cuntz equivalence. []
Definition 1.6. A positive contraction a is RICH if for every
tau >= 0 the cutoff (a - tau)_+ is properly infinite.
Zero is rich. Under (P) every positive contraction is rich, since
each scalar cutoff is properly infinite by hypothesis.
A RAMP for a positive contraction a is a continuous nondecreasing
function f: [0,1] -> [0,1] that is zero on a neighbourhood of 0 and
equal to 1 beyond some specified larger threshold. (Rampless
cutoffs use (t)_+ = max(t,0); ramps are the smooth versions that
keep a spectral GAP near 0 and a PLATEAU at 1.)
Lemma 1.7 (ramps of rich elements are rich, with the signal). Let a
be a rich positive contraction and f a ramp for a. Then f(a) is a
rich positive contraction in Her(a); and if a representation pi has
||pi(a)|| = 1 on a set where a >= 1 (in particular whenever a is a
contraction with ||pi(a)|| > 1/2 and f = 1 on [1/2,1]), then
||pi(f(a))|| = 1. More generally, for 0 <= tau < 1 the open
positivity set of (f(a) - tau)_+ is either empty or the positivity
set of (a - rho)_+ for some rho >= 0.
Proof. For 0 <= tau < 1, f(a) >= tau forces a >= rho for the
smallest rho with f(rho) = tau (f nondecreasing, f = 1 at the top).
So (f(a) - tau)_+ and (a - rho)_+ have the same open positivity set;
Lemma 1.5(iii) gives the same hereditary algebra and Cuntz class, and
richness of a makes (a - rho)_+ properly infinite, hence (f(a) -
tau)_+ properly infinite. For tau >= 1 the cutoff is 0. The
spectral mapping clause is immediate from the plateau f = 1 at 1 and
the fact that ||pi(a)|| = 1 forces the spectral projection of a above
1 - eta to be nonzero for each eta. []
Two facts about Prim(A) are used in route E. Recall from 0.4:
Omega(c, alpha) = {P : ||c_P|| > alpha} (open) and
K(c, alpha) = {P : ||c_P|| >= alpha}.
Lemma 1.8 (pure-state norm attainment). If B is a Cstar-algebra and
0 != b in B_+, there is an irreducible representation pi of B with
||pi(b)|| = ||b||.
Proof. In the state space of the unitization B~, the set
F_b = {phi : phi(b) = ||b||} is a nonempty weak-star compact face
(evaluation at the top spectral value on C*(1,b) extends by the
positive Hahn-Banach theorem; every state has value <= ||b|| on b).
By Krein-Milman it has an extreme point, which is a pure state of B~;
its GNS representation is irreducible. It does not annihilate B
(value on b is positive), and restricting it to B is irreducible and
nondegenerate, with ||pi(b)|| >= ||b||, hence equal by
contractivity. []
Lemma 1.9 (primitive norm slices). Let A be a Cstar-algebra. For
c in A_+ and alpha > 0 the set Omega(c, alpha) is open and K(c, alpha)
is quasi-compact. No Hausdorff hypothesis is required.
Proof. Omega(c, alpha) = {P : (c - alpha)_+ is not in P}, open in
the hull-kernel topology. For K(c, alpha): take a relative open
cover and extend to ambient open sets O_lambda = {P : J_lambda not in
P}, where J_lambda are closed ideals. Put I = closure of
sum_lambda J_lambda. If ||c + I|| >= alpha, Lemma 1.8 applied in A/I
gives an irreducible representation attaining that norm; its
pullback has primitive kernel containing I, lies in K(c, alpha), and
lies outside every O_lambda -- contradicting the covering property.
Hence ||c + I|| < alpha (this is also clear if A/I = 0). Choose d in
I with ||c - d|| < alpha, then approximate d by d_F in sum over a
finite F of J_lambda closely enough that ||c - d_F|| < alpha still
holds. Any primitive ideal containing every J_lambda for lambda in F
has ||c_P|| < alpha, so the finitely many O_lambda for lambda in F
cover K(c, alpha). The proof uses quasi-compactness only and does
not assert K(c, alpha) is closed. []
Throughout, C is a Cstar-algebra, C~ is its unitization
(C~ = C when C is unital), and the unitization acts canonically on
C**. Fix once and for all
s = 1/4, alpha_0 = 1/32,
and, for a chosen integer k >= 1 (the number of graph entries), fix
positive reals gamma_g, sigma_g with
gamma_g^2 + k * sigma_g^2 < 1. (GC)
For example gamma_g = 1/2, sigma_g = 1/(2 sqrt(k+1)) work. Both k
and the coefficients are FIXED BEFORE the measurements below are
computed.
The goal of this part. Given positive contractions a, y in C with a
rich, and a contraction c in C~, produce positive contractions
p_1 in Her(a), q_1 in Her(y), with p_1 rich, such that:
SIGNAL for every representation pi of C,
||pi(a)|| > 1/2 ==> ||pi(p_1)|| = 1,
||pi(y)|| > 1/2 ==> ||pi(q_1)|| = 1;
SUPPORT-BOUND in the universal bidual C**,
|| e_{p_1} c e_{q_1} || < 1.
The support bound is what later, by iteration, gives arbitrarily small
mixed compression. It comes in two steps: a GRAPH turns failure of
the bound into finitely many explicit measurement equalities, and a
NONVANISHING implication plus a cutoff converts failure into a strict
loss of norm. The finite exclusion of those equalities is the
job of the four-entry step (under (P)) and of the nuclear step.
Proposition 2.1 (fixed measurements of a graph). Let C be a
Cstar-algebra, a, y in C_+ contractions with a rich, c in C~ a
contraction. For an integer k >= 1 and fixed gamma_g, sigma_g
satisfying (GC), there exist
X in Her((a - 1/8)_+)_+, B_1, ..., B_k in Her((a - 1/8)_+),
with the following property. For ANY contractions
S_1, ..., S_k in Her((X - alpha_0)_+), one can form a rich positive
contraction p in Her(a) such that, for every representation pi of C,
||pi(a)|| > 1/2 implies ||pi(p)|| = 1; and if a representation pi
satisfies
pi( y (1 - c* p^2 c) y ) = 0 and ||pi(y)|| > s,
then, in that same representation,
||pi(X)|| > s, pi(B_i) = pi(S_i) (1 <= i <= k). (GF)
The data X, B_1, ..., B_k are FIXED BEFORE the S_i are chosen.
Idea. Build k+2 orthogonal "slots" inside a cutoff of a (a rich a
contains arbitrarily many orthogonal copies of its high part). Spread
a prospective isometry L over the slots with weights gamma_g, sigma_g
and the freely chosen entries S_i, and set p = L t L*, t a ramp of a.
If the mixed defect y(1 - c* p^2 c)y vanishes and y is detected, then
equality in contraction inequalities forces c to be an isometry on the
support of y, which forces F = c y c* to live on the p = 1 space, i.e.
on the range of L pulled back to the t = 1 space. Reading the
coordinates off that form both computes X (from the 0-slot) and
reveals each S_i (from the i-slot); the measurements were designed so
that these readings equal the fixed X and B_i.
Proof. Let e = e_(a - 1/8)_+ and H = Her((a - 1/8)_+). Because a is
rich, (a - 1/8)_+ is properly infinite, so
a^(+(k+2)) precsim a. Apply the transport Lemma 1.3 with cutoff 1/8
and split the resulting row into k+2 slots labelled 0, 1, ..., k, *.
There are v_0, v_1, ..., v_k, v_* in C** with
v_i* v_j = delta_ij e, v_i v_i* <= e_a, v_i H in C. (SL)
Choose a ramp t = t(a) in H that is 0 on [0,1/4] and 1 on [1/2,1]; it
is rich (Lemma 1.7) and its represented norm is 1 wherever the
represented norm of a exceeds 1/2.
Compute the measurements now, before choosing any S_i. Put
F = c y c* in C_+. The design calculation: if L in C** satisfies
L*L = e with slot coordinates v_0* L = gamma_g e, v_i* L = sigma_g
S_i, and F = L H_0 L* for a positive H_0 in e C** e, then
v_0* F v_0 = gamma_g^2 H_0, v_i* F v_0 = sigma_g gamma_g S_i H_0.
So the 0-slot recovers H_0, and an inverse on its spectral cutoff
recovers each S_i from the i-slot. Choose a continuous nonnegative
function h_inv on [0,inf), zero at 0, equal to 1/lambda for
lambda >= alpha_0, bounded by 1/alpha_0. Define
X = (1/gamma_g^2) t v_0* F v_0 t,
B_i = (1/(sigma_g gamma_g)) t v_i* F v_0 t h_inv(X). (GM)
The sandwiches are (v_i t)* F (v_0 t), hence lie in C; they are
supported on both sides by e, so lie in H; the same holds after
multiplication by h_inv(X) (a function of X in H). Thus X >= 0 and
X, B_i are in H, and the slots, t, c, y fix them WITHOUT using any S_i.
Now fix the S_i in Her((X - alpha_0)_+). In the corner e C** e set
J = ( (1 - gamma_g^2) e - sigma_g^2 sum_{i=1}^k S_i* S_i )^(1/2),
L = gamma_g v_0 + sigma_g sum_{i=1}^k v_i S_i + v_* J,
p = L t L*.
The radicand is at least (1 - gamma_g^2 - k sigma_g^2) e > 0; its
square root lies in H + C e and multiplies H into itself, so L h in C
for every h in H (term by term in (SL)). The slot identities give
L* L = e, e_a L = L.
By the membership-and-support argument of Lemma 1.3, h -> L h L* is an
injective *-homomorphism from H into Her(a): for h >= 0 its value is
(L h^(1/2))(L h^(1/2))* in C, supported under e_a, and L*L = e gives
multiplicativity. The same identities preserve cutoffs, their Cuntz
classes, and represented norms, exactly as in Lemma 1.3. Hence p is a
rich positive contraction in Her(a), and ||pi(a)|| > 1/2 gives
||pi(t)|| = 1, whence ||pi(p)|| = ||pi(L t L*)|| = 1. (If e = 0, all
corner operators are 0 and the assertions are vacuous.)
It remains to read off what vanishing of the defect forces. Work on
the essential subspace of a representation in which
d = y (1 - c* p^2 c) y = 0, ||y|| > s,
and drop the representation symbols. For every vector xi,
||y xi|| = ||p c y xi|| <= ||c y xi|| <= ||y xi||.
Equality first on the range of y, then on its closure, shows c is
isometric on e_y H. For a positive contraction p, "||p zeta|| =
||zeta||" implies "p zeta = zeta" (apply 1 - p^2). So c(e_y H) lies
in the p = 1 subspace, and F = c y c* is supported on that subspace.
Since p = L t L* with L*L = e, the p = 1 subspace lies in the range of
L, and its pullback lies in the t = 1 subspace. Thus, with
H_0 = L* F L,
F = L H_0 L*, t H_0 = H_0 = H_0 t, ||H_0|| = ||F|| = ||y|| > s.
(The last equality uses isometric action of c on e_y H.)
Now v_0* L = gamma_g e, v_i* L = sigma_g S_i, L* v_0 = gamma_g e.
Substituting in (GM):
X = t H_0 t = H_0,
B_i = t S_i H_0 t h_inv(X).
Since t X = X, t is the identity on the closure of the range of X =
e_X H; each represented S_i is supported under e_(X - alpha_0)_+, so
t S_i = S_i and S_i X h_inv(X) = S_i. The display gives B_i = S_i,
while ||X|| = ||H_0|| = ||y|| > s. This is (GF). []
So it suffices to choose the S_i inside their spectral corner so that
the relations (GF) CANNOT all hold. The graph then supplies the
NONVANISHING implication that the next lemma assumes.
Lemma 2.2 (from nonvanishing to shrinking). Let C be a Cstar-algebra,
a, y in C_+ contractions with a rich, c in C~ a contraction. Suppose
p in Her(a) is a rich positive contraction with
||pi(a)|| > 1/2 ==> ||pi(p)|| = 1 for every representation pi.
Put d = y (1 - c* p^2 c) y in C_+, and assume
pi(d) = 0 ==> ||pi(y)|| <= 1/4 for every representation pi.
(NV)
Then there are positive contractions p_1 in Her(a), q_1 in Her(y) such
that p_1 is rich, with the two signals
||pi(a)|| > 1/2 ==> ||pi(p_1)|| = 1, ||pi(y)|| > 1/2 ==> ||pi(q_1)|| = 1,
for every representation pi, and the single universal-bidual bound
|| e_{p_1} c e_{q_1} || < 1.
Idea. (NV) says the ideal of d already contains the "y above 1/4"
cutoff. So a cutoff of d controls a cutoff of y; compare the positive
q = y h_d y (which is Cuntz equivalent to h_d) into the y-corner,
keeping a norm signal. On the support where d >= delta, the defect
identity ||p c y xi||^2 = ||y xi||^2 - <d xi, xi> loses a FIXED
fraction of norm, so ||p c e_q|| <= sqrt(1 - delta) < 1; a ramp p_1 of
p on a high spectral window then keeps the a-signal while making the
support contraction strictly below 1.
Proof. The contractions give 0 <= d <= y^2 <= 1. By (NV), every
irreducible representation vanishing on d vanishes on (y - s)_+ with
s = 1/4. The primitive ideal test (Lemma 1.5(ii)) gives
(y - s)_+ in Id(d).
First ideal step. Set s' = 3/8. Apply cutoff exhaustion
(Lemma 1.5(i)) to (y - s)_+ with cutoff s' - s = 1/8: after possibly
decreasing a parameter, there is 0 < delta < 1 with
(y - s')_+ in Id(h_d), h_d = (d - delta)_+. (I1)
Define z = y h_d^(1/2) in C, q = z z* = y h_d y in Her(y). The order
0 <= d <= y^2 <= 1 gives
delta h_d <= h_d^(1/2) d h_d^(1/2) <= z*z = h_d^(1/2) y^2 h_d^(1/2)
<= h_d. (ORD)
So z*z and h_d have the same hereditary algebra, and z*z ~_Cu zz* = q
(Lemma 1.2(ii)); hence Id(q) = Id(h_d). A second cutoff exhaustion, to
(y - s')_+ with cutoff 1/2 - s' = 1/8, gives delta' > 0 with
(y - 1/2)_+ in Id((q - delta')_+). (I2)
Let q_1 be a ramp of q, zero on [0, delta'/2] and 1 on [delta', inf).
It is a positive contraction in Her(q) in Her(y). If ||pi(y)|| > 1/2
then pi((y - 1/2)_+) != 0; by (I2) pi((q - delta')_+) != 0, so
||pi(q)|| > delta' and ||pi(q_1)|| = 1. This is the y-signal.
Universal-bidual norm bound. Work in the universal nondegenerate
representation. For xi in e_{h_d} H = 1_(delta,inf)(d) H,
||p c y xi||^2 = ||y xi||^2 - <d xi, xi>
<= ||y xi||^2 - delta ||xi||^2
<= (1 - delta) ||y xi||^2. (NL)
(The last uses y a contraction.) Apply (NL) to vectors in the range
of h_d^(1/2); the closure of the range of y h_d^(1/2) is e_q H, so by
continuity ||p c e_q|| <= sqrt(1 - delta). Choose
sqrt(1 - delta) < tau < 1, and let p_1 be a ramp of p, zero on [0,tau]
and 1 at 1. Functional calculus gives e_{p_1} <= tau^(-2) p^2, while
e_{q_1} <= e_q. Hence
|| e_{p_1} c e_{q_1} ||
<= tau^(-1) || p c e_{q_1} ||
<= sqrt(1 - delta) / tau < 1.
The ramp p_1 of the rich p stays rich and in Her(a) (Lemma 1.7), and by
spectral mapping has norm 1 in every representation where p does. This
proves the a-signal; the y-signal was proved above. Zero inputs are
covered. []
Throughout this part C satisfies (P): EVERY positive element is
properly infinite. Pick four freely chosen entries that rule out
the fixed measurement relations (GF) of Proposition 2.1. This feeds
the nonvanishing hypothesis (NV) of Lemma 2.2 and hence gives the
strict support contraction. A SPECTRAL CUTOFF is essential: it puts
a support into an ACTUAL operator range, where the positivity
argument of Lemma 3.1 applies.
Constants (fixed, increasing in absolute value as shown):
alpha_0 = 1/32 < alpha = 1/16 < beta = 1/8 < s = 1/4,
h = 1/2.
Lemma 3.1. Let x, Q be bounded positive operators on a Hilbert space
H, and put U = x Q^2 x. For every eta > 0,
e_(U - eta)_+ H is contained in ran(x Q), (AR)
ker Q intersected with ran(x Q) = {0}.
Proof. Let f_eta be the bounded Borel function equal to 1/lambda for
lambda > eta and 0 otherwise; its sup on the spectrum is at most
1/eta. If zeta in e_(U - eta)_+ H, then
zeta = U f_eta(U) zeta = x Q ( Q x f_eta(U) zeta ),
exhibiting zeta in ran(x Q). If zeta = x Q xi and Q zeta = 0, then
0 = <Q x Q xi, xi> = || x^(1/2) Q xi ||^2,
so zeta = x Q xi = 0. The statement is about the ACTUAL range; no
closed-range conclusion is used. []
Proposition 3.2. Suppose C satisfies (P). Let X in C_+ and
B_1, B_2, B_3, B_4 in C. Then there exist contractions
T_1, ..., T_4 in Her((X - alpha_0)_+) such that NO representation pi
of C satisfies
||pi(X)|| > s, pi(B_i) = pi(T_i) (1 <= i <= 4). (4F)
Idea of the contradiction. Suppose every choice of four entries could
be matched in some representation. Then, testing only finitely many
fixed elements at a time and letting a slack eta decrease, all the
data pass to a NORM ULTRAPRODUCT. In an irreducible representation of
the resulting image algebra:
* the FIRST PAIR behaves as two orthogonal projections forcing an
ISOMETRY v = r_1 + b_0 that has a NONZERO defect projection P
(the Blackadar-Cuntz scaling element): r_1, r_2 are two
orthonormal isometries pushed below the B_1-corner, and b_0 is the
"complement" of the B_1-corner, chosen so that v is an isometry
orthogonal to r_2;
* the LAST PAIR is designed from a FIXED formula P_0 that, once the
defect P is present, EQUALS P. Then B_3 B_3* is forced to act as h
on P (from the relation R = 0) while B_3 kills the support
h_P(P_0) B_4 = B_4 on the OTHER side; since B_3* B_4 = 0, this
forces h B_4 = 0 and then B_4 = B_3 = 0, contradicting h > 0 on
P != 0.
Everything varying (the T_i, the finite test sets, the slacks) is
discarded in the ultraproduct; only FIXED elements and FIXED
functional-calculus expressions survive. The directed index set,
rather than a countable subsequence, is what makes this work with NO
separability assumption.
Proof. Suppose, for contradiction, that every FOUR-TUPLE of
contractions in Her((X - alpha_0)_+) can be matched, i.e. there is
some representation satisfying (4F). Build the derivation of a
contradiction.
Fixed elements. Define
x = (X - alpha)_+, V = (X - beta)_+, Q = (h 1 - B_1 B_1*)_+,
U = x Q^2 x. (FX)
Here Q lives in C~ (the unit 1 is the unitization's), while
x, V, U live in C_+.
Index set and slack. Let Fin(C) be the set of finite subsets of C and
order
Lambda = N_{>=1} x Fin(C)
by the usual order in the first coordinate and inclusion in the
second. At lambda = (m, E), put eta = 1/m, so eta -> 0 along this
directed set, and every fixed w in C eventually lies in E.
FIRST PAIR. The first pair is engineered so that, in the limiting
image, U w V = 0 for ALL fixed w in C. At index (m, E) test only the
finitely many w in E; one positive sum controls them all:
A_eta = (U - eta)_+,
D_0 = A_eta ( sum_{w in E} w V w* ) A_eta,
D = g_eta(D_0), (D0)
where g_eta is a ramp zero on [0, eta] and 1 on [2 eta, inf). Indeed
D_0 = sum_{w in E} (A_eta w V^(1/2)) (A_eta w V^(1/2))*,
so a small norm for D_0 makes every A_eta w V^(1/2) small. Also
D_0 in Her(A_eta) intersected with Id(V),
e_D <= e_(D_0 - eta)_+ <= e_{A_eta} <= e_x.
Since V is properly infinite (this is (P)!), ideal comparison
(Lemma 1.2(iii)) gives D_0 (+), D_0 precsim V. Apply transport
(Lemma 1.3) with cutoff eta, split the carrier into two columns, and
multiply these columns by D^(1/2). Taking adjoints of the resulting
two elements gives contractions T_1, T_2 in C with
T_i T_j* = delta_ij D, e_V T_i* = T_i* (i, j = 1, 2). (T12)
Their left supports lie under e_D and their right supports under e_V;
both are under e_x, hence under e_(X - alpha_0)_+, so they are in the
required hereditary algebra.
Now take a representation satisfying (4F) at this index and drop the
symbols. Then B_1 B_1* = D and Q = (h 1 - D)_+. The spectral
subspace of D for (h, inf) is killed by Q and lies in
e_D H contained in e_(U - eta)_+ H. Lemma 3.1 (actual range) says this
subspace is 0. Hence
B_1 B_1* = D <= h 1, ||D_0|| <= 2 eta in this rep. (B1)
For the second inequality: were ||D_0|| >= 2 eta, spectral mapping and
the plateau of g_eta would give ||D|| = 1 > h.
A FIXED PROSPECTIVE DEFECT HANDED TO THE LAST PAIR. Choose ramps
H = h_0(X), K = k_0(X)
with h_0 vanishing near [0, alpha] and 1 on [beta, inf); k_0 vanishing
near [0, alpha_0] and 1 on [alpha, inf). In particular K H = H. Set
r_i = h^(-1/2) B_i* H (i = 1, 2), b_0 = (1 - H^2)^(1/2). (r)
In the limiting irreducible representation below, the first pair will
give the relations
r_i* r_j = delta_ij H^2, H r_i = r_i, H != 0,
and these imply b_0 r_i = 0 and so
v = r_1 + b_0, v* v = H^2 + b_0^2 = 1, v* r_2 = 0.
Since r_2* r_2 = H^2 != 0, the isometry v has a NONZERO defect
projection 1 - v v*. (This is the Blackadar-Cuntz scaling element; the
second entry supplies the nonzero orthogonal range r_2.) None of this
is asserted in C yet; the relations hold only in the limit.
Encode this prospective defect by ONE FIXED element of C, chosen
before any limit. Compression by K keeps it in the required
hereditary algebra, and clipping makes it a positive contraction even
before the relations hold. With chi(t) = min(1, max(0, t)) on R,
P_0 = chi( K [1 - (r_1 + b_0)(r_1 + b_0)*] K ) in C. (P0)
The compressed selfadjoint is in C and supported under e_K; since
chi(0) = 0, functional calculus in Her(K) gives
e_{P_0} <= e_K <= e_(X - alpha_0)_+.
This membership and support use NO relation from any representation.
LAST PAIR. Define the additional fixed elements
x' = (P_0 - 1/2)_+, Q' = (h 1 - B_3 B_3*)_+, R = x' (Q')^2 x'. (LF)
At the SAME index eta, property (P) gives
((R - eta)_+)^(+2) precsim (R - eta)_+.
Apply transport (Lemma 1.3) to this comparison with cutoff eta. If
t_R is a ramp of R, zero on [0, 2 eta] and 1 on [4 eta, inf), the two
carrier columns times t_R give contractions T_3, T_4 in C with
T_3* T_4 = 0, T_3* T_3 = T_4* T_4 = t_R^2. (T34)
Their left supports lie under e_(R - eta)_+ and their right supports
under e_(R - 2 eta)_+; both are under e_{x'} in e_(X - alpha_0)_+.
Transport preserves represented norms, so
||pi(R)|| > 4 eta ==> ||pi(T_3)|| = 1. (Tn)
In a representation satisfying (4F) at this index, B_3 = T_3. Apply
Lemma 3.1 (actual range) to x', Q', R. The spectral subspace of
B_3 B_3* above h is killed by Q' and lies under e_(R - eta)_+, so it
is zero. Then (Tn) gives
B_3 B_3* <= h 1, ||R|| <= 4 eta in this rep. (B3)
PASSING ONLY FIXED ELEMENTS TO AN ULTRAPRODUCT. Suppose, for a
contradiction, that EVERY index of Lambda admits a representation
satisfying (4F). Choose one, called pi_lambda, on its nondegenerate
essential subspace. (No size issue: from any such representation,
pick a vector detecting ||pi_lambda(X)|| > s and restrict to the
reducing cyclic subrepresentation; a set of cyclic reps suffices.)
Choose an ultrafilter omega containing all tails of Lambda, and form
the norm ultraproduct
Bext = prod_omega B(H_lambda)
= ( prod_lambda B(H_lambda) ) / { (z_lambda) :
lim_omega ||z_lambda|| = 0 }.
The norm of a class is lim_omega ||z_lambda||. Let
Phi: C -> Bext, Phi(z) = [(pi_lambda(z))], and write B0 = Phi(C)
(a closed C*-subalgebra). Unital extensions carry the unitized
formulas into C*(B0, 1). Suppress Phi on fixed elements.
For any FIXED w in C, eventually w in E. Positivity of the summands
in (D0) and (B1) gives, on that tail,
|| pi_lambda( (U - eta)_+ w V^(1/2) ) ||^2 <= 2 eta.
Since ||U - (U - eta)_+|| <= eta,
|| pi_lambda( U w V ) ||
<= sqrt(2 eta) ||V||^(1/2) + eta ||w|| ||V|| -> 0. (UCV)
Hence U B0 V = 0 in the image algebra. The other exact relations and
bounds at each index give
||X|| >= s,
B_1 B_1* <= h 1,
B_i B_j* = delta_ij B_1 B_1*, H B_i* = B_i* (i,j=1,2),
U B0 V = 0,
R = 0,
B_3 B_3* <= h 1,
B_3* B_4 = 0, B_3* B_3 = B_4* B_4. (UP)
For "H B_i* = B_i*" use e_V T_i* = T_i* and H = 1 on the support of V.
Fix also a continuous h_P: [0,1] -> [0,1] with h_P(0) = 0 and
h_P = 1 on [1/2,1]. The left support of T_4 lies under e_{x'}, so
h_P(P_0) B_4 = B_4. (B4s)
Every element in (UP) and (B4s) is the image of a fixed algebra element
or a fixed unitized functional-calculus expression. We have NOT put
the varying T_i, D, D_0, t_R into B0, and we do NOT require proper
infiniteness of B0 or of Bext. The directed finite sets in (UCV)
handle all w in C without separability.
A NONZERO DEFECT GIVES THE CONTRADICTION. Since ||X|| >= s > beta,
the image V = (X - beta)_+ is nonzero. Choose an irreducible
representation of B0 in which V != 0. Extend it to
C*(B0, 1) = B0 + C 1 by sending 1 to the identity (valid even if B0 has
a different unit). The unitized expressions Q, b_0, Q' and the defect
below then have their stated meanings; suppress the representation
symbols again. The closed span of B0 V H is a nonzero reducing
subspace, hence all of H. So U B0 V = 0 forces U = 0.
Put D_1 = B_1 B_1*. Since D_1 <= h 1, Q = h 1 - D_1, and
0 = U = (Q x)*(Q x) => D_1 x = h x.
By closure D_1 = h 1 on e_x H. The support of H lies there. From
(r) and (UP):
r_i* r_j = delta_ij H^2, H r_i = r_i (i,j=1,2). (ISO)
Also H V = V != 0, so H != 0 and r_2 != 0.
The conditional calculation before (P0) now applies: v = r_1 + b_0 is
an isometry with v* r_2 = 0. Its defect projection
P = 1 - v v*
is nonzero, since P r_2 = r_2 != 0.
CHECK P_0 = P (this is where BOTH sides of the cutoff matter). The
scalar ramps give K H = H. From H r_1 = r_1 follows K r_1 = r_1, and
r_1 = h^(-1/2) B_1* H gives r_1 K = r_1. Also K commutes with b_0.
Expanding,
P = H^2 - r_1 r_1* - r_1 b_0 - b_0 r_1*,
and each term is fixed by multiplication by K on both sides.
Therefore K P K = P, and chi(P) = P (P a projection). Formula (P0)
gives
P_0 = P. (PP)
Now x' = (P_0 - 1/2)_+ = P/2. Put D_3 = B_3 B_3* <= h 1; then
Q' = h 1 - D_3. The relation R = 0 gives Q' x' = 0, so
D_3 P = h P. (LPC)
Equations (B4s) and (PP) say B_4 = P B_4. But B_3* B_4 = 0, so
D_3 B_4 = 0, whereas (LPC) gives D_3 B_4 = h B_4. Thus B_4 = 0, and
B_3* B_3 = B_4* B_4 gives B_3 = 0. That contradicts (LPC), since
h > 0 and P != 0.
So some index has NO representation satisfying (4F); the four entries
chosen at that index prove Proposition 3.2. []
Assembling Parts 2 and 3.
Proposition 3.3 (shrinking under individual proper infiniteness).
Suppose C satisfies (P). Let a, y in C be positive contractions and
c in C~ a contraction. Then there are positive contractions
p_1 in Her(a), q_1 in Her(y), with p_1 rich, such that for every
representation pi of C,
||pi(a)|| > 1/2 ==> ||pi(p_1)|| = 1,
||pi(y)|| > 1/2 ==> ||pi(q_1)|| = 1, (SP-sig)
and, in the universal bidual,
|| e_{p_1} c e_{q_1} || < 1. (SP-sup)
Proof. Under (P) every positive contraction is rich (each scalar
cutoff is properly infinite). In Proposition 2.1 (the graph) fix
k = 4 BEFORE the measurements are computed, and choose, e.g.,
gamma_g = 1/2, sigma_g = 1/(2 sqrt 5); then
gamma_g^2 + 4 sigma_g^2 < 1. The graph gives fixed X and
B_1, ..., B_4 for the inputs a, y, c. Apply Proposition 3.2 (four
entries) to these and use its contractions T_i as the graph entries
S_i. The resulting rich positive contraction p in Her(a) has the
required norm-one signal for a.
Put d = y (1 - c* p^2 c) y. If a representation pi had pi(d) = 0 and
||pi(y)|| > 1/4, the graph implication (GF) would give ||pi(X)|| > 1/4
and pi(B_i) = pi(T_i) for every i. These are exactly the relations
excluded by Proposition 3.2. So
pi( y (1 - c* p^2 c) y ) = 0 ==> ||pi(y)|| <= 1/4 (NV-P)
for every representation. The same p is rich and carries the
a-signal, so Lemma 2.2 applies to a, y, c, p and gives the stated
p_1, q_1. Zero inputs are covered. []
Throughout this part C is a Cstar-algebra satisfying (P). Everything
below is proved; the only "external input" is the definition of (P).
Lemma 4.1. Let (C_i)_{i in I} be ANY family of Cstar-algebras
satisfying (P), with bounded product D = prod_{i in I} C_i (sup
norm). Then D satisfies (P). More precisely, for a positive
contraction d = (d_i) in D and 0 < delta < 1/2 there is a row
s = (s_i) in M_(1,2)(D) such that
s* d s = (d - 2 delta)_+ (+) (d - 2 delta)_+, ||s|| <= delta^(-1/2). (PR)
Proof. If I is empty then D = 0 and the zero row works. Otherwise,
for each i, put
h_i = (d_i - delta)_+, k_i = (d_i - 2 delta)_+ (+) (d_i - 2 delta)_+.
The positive h_i is properly infinite (each cutoff of a (P)-algebra).
Apply the transport Lemma 1.3 to h_i (+) h_i precsim h_i at cutoff
delta. The resulting rectangular partial isometry v_i in M_(1,2)(C_i)
satisfies v_i k_i^(1/2) = r_i with
r_i* r_i = k_i, e_{h_i} r_i = r_i, ||r_i|| <= 1.
Define phi_delta on [0,1] by
phi_delta(t) = t / delta^(3/2) for 0 <= t <= delta,
phi_delta(t) = t^(-1/2) for t >= delta.
This is continuous, vanishes at 0, bounded by delta^(-1/2), and
satisfies t^(1/2) phi_delta(t) = 1 for t > delta. Set
s_i = phi_delta(d_i) r_i in M_(1,2)(C_i).
Then d_i^(1/2) s_i = r_i (because phi_delta inverts d_i^(1/2) on the
support of h_i) and ||s_i|| <= delta^(-1/2). The uniform bound places
(s_i) in M_(1,2)(D); coordinatewise functional calculus gives (PR).
Its right-hand side converges to d (+) d in norm as delta down to 0,
whence s* d s = d (+) d in the limit, so d is properly infinite.
Scaling covers every positive element of D. []
(The product-comparison criterion of Antoine-Perera-Thiel also gives
this permanence; the rows above are the concrete witnesses.)
Lemma 4.2 (uniform support contraction). There exists a UNIVERSAL
constant theta in (0,1) such that, for every Cstar-algebra C
satisfying (P), every pair of positive contractions a, y in C and
every contraction c in C, there are positive contractions p in
Her(a), q in Her(y) with
||pi(a)|| > 1/2 ==> ||pi(p)|| = 1,
||pi(y)|| > 1/2 ==> ||pi(q)|| = 1, (US-sig)
for every representation pi of C, and
|| e_p c e_q || <= theta. (US-sup)
Proof (by contradiction). If no universal bound existed, then for
each integer m >= 2 we could choose a (P)-algebra C_m and positive
contractions a_m, y_m, c_m (with c_m a contraction) such that EVERY
pair with the stated hereditary membership and signals has
|| e_{p_m} c_m e_{q_m} || > 1 - 1/m.
Form the bounded product D = prod C_m and its coordinate three
a = (a_m), y = (y_m), c = (c_m) in D. Lemma 4.1 gives (P) for D.
Now apply Proposition 3.3 (shrinking under (P)) inside D, obtaining
p, q with the signals and
kappa = || e_p c e_q || < 1.
The coordinate *-homomorphism pi_m: D -> C_m is unital and contractive,
and its normal extension sends e_p to e_{p_m}, e_q to e_{q_m}. Hence
the coordinate support norm ||e_{p_m} c_m e_{q_m}|| <= kappa. Also,
if pi is any representation of C_m, its composition pi o pi_m is a
representation of D, so both signals hold for the coordinate pair.
Thus the pair (p_m, q_m) at coordinate m satisfies both signals AND
has support norm <= kappa, contradicting the strict lower bound
chosen at m when 1 - 1/m > kappa. Therefore a universal theta < 1
exists. []
(The input c belongs to C, not only C~; the sequence c = (c_m) must
lie in the algebra D. A sequence of arbitrary individual unitizations
need not. This is why the statement uses contractions of C.)
Proposition 4.3 (iterated separation). Suppose C satisfies (P). For
positive contractions a_*, b_* in C, a contraction c_* in C, and
zeta > 0, there are positive contractions p in Her(a_*), q in Her(b_*)
with
|| e_p c_* e_q || < zeta, (IS-sup)
and, for every representation pi of C,
||pi(a_*)|| > 1/2 ==> ||pi(p)|| = 1,
||pi(b_*)|| > 1/2 ==> ||pi(q)|| = 1. (IS-sig)
Proof. Fix theta from Lemma 4.2. On [0,1] set
f(t) = (2t - 1)_+, g(t) = min(1, (4t - 1)_+).
Both vanish near 0; f(1) = 1; g = 1 on [1/2, 1]. For a positive
contraction h,
g(h) = e_h g(h), g(h) e_{f(h)} = e_{f(h)}. (Icut)
Apply Lemma 4.2 once to a_*, b_*, c_*, obtaining p_1, q_1 with
support norm <= theta. Suppose inductively that p_k in Her(a_*),
q_k in Her(b_*) are positive contractions with both signals and
|| e_{p_k} c_* e_{q_k} || <= theta^k.
Normalize the next input. By (Icut) the normalized element
c^(k) = theta^(-k) g(p_k) c_* g(q_k) in C
has norm <= 1: g(p_k) is supported under e_{p_k}, so g(p_k) c_* g(q_k)
is e_{p_k} c_* e_{q_k} in disguise (multiplied by 0 <= g <= 1 on both
sides), whose norm is at most theta^k. Apply Lemma 4.2 to
f(p_k), f(q_k), c^(k) and call the outputs p_(k+1), q_(k+1). The
new supports lie under those of f(p_k), f(q_k); by (Icut),
e_{p_(k+1)} c_* e_{q_(k+1)} = theta^k e_{p_(k+1)} c^(k) e_{q_(k+1)},
so
|| e_{p_(k+1)} c_* e_{q_(k+1)} || <= theta^(k+1).
The hereditary algebras are nested: Her(p_(k+1)) is contained in
Her(f(p_k)) is contained in Her(p_k) is contained in Her(a_*); and
the same for q. In every representation, ||pi(p_k)|| = 1 implies
||pi(f(p_k))|| = 1 (spectral mapping, f(1) = 1), which in turn, by
the uniform lemma's signal, implies ||pi(p_(k+1))|| = 1. The
b_*-signal persists the same way. Finally choose k with
theta^k < zeta, and put p = p_k, q = q_k. []
Proposition 4.4 (bounded diagonalization). Suppose C satisfies (P).
Let a, b in C_+ be positive contractions, c in C a contraction,
0 < delta < 1/2, 0 < rho < 2 delta, and zeta > 0. Then there are
elements s, t in C with
||s|| , ||t|| <= delta^(-1/2),
||s* a s - a|| <= 2 delta + rho,
||t* b t - b|| <= 2 delta + rho, (BDbnd)
||s* c t|| < zeta.
Proof. Choose ramps f, F on [0,1] such that
f = 0 on [0, delta/2], f = 1 on [delta, 1],
F = 0 on [0, delta], F = 1 on [2 delta, 1],
both continuous with 0 <= f <= F <= 1, f F = f. (Extending linearly
gives such functions.) Put
a_* = F(a), b_* = F(b).
Both are positive contractions: a_* in Her(a), b_* in Her(b), since F
vanishes at 0. Also, e_{a_*} <= 1_{(delta, inf)}(a), the support of
(a - delta)_+.
Choose zeta' = delta * zeta. Apply Proposition 4.3 (iterated
separation) to a_*, b_*, c with target zeta', getting positive
contractions p in Her(a_*), q in Her(b_*), with the two signals
and
|| e_p c e_q || < zeta' = delta * zeta. (BD-eps)
Because e_p <= e_{a_*} <= 1_{(delta, inf)}(a), on the range of p the
element a exceeds delta, so
p a p >= delta p^2. (BD-pa)
Consequently Id(pap) = Id(p): one inclusion is immediate (pap in
Her(p)); the other follows from (BD-pa) (p^2 generates Id(p), so
Id(p) is contained in Id(pap)).
Put d_a = (a - 2 delta)_+. CLAIM: d_a in Id(p). If an irreducible
representation pi detects d_a, ||pi(a)|| > 2 delta, then F(a) has
norm one in that representation (F = 1 on [2 delta, 1]), so
||pi(a_*)|| = 1 > 1/2, the signal gives ||pi(p)|| = 1, and
||pi(pap)|| >= delta pi(p)^2 != 0. Hence d_a in Id(pap) = Id(p).
By Lemma 1.5(ii) (primitive ideal test), d_a in Id(p).
The positive pap is properly infinite under (P), so by
Lemma 1.2(iii), in the common stabilization,
d_a precsim pap. (BD-cmp)
Compressing a stabilization comparison coefficient to the scalar
source and target corners supplies w in C with
|| w* pap w - d_a || < rho. (BD-w)
(Here d_a is scalar; at matrix level use zero-padded coefficients
and reduce back, which is legitimate because both sides lie in the
scalar corner.) Set s = pw. Then e_p s = s (since pap >= delta p^2
gives p(something)s = s ... more explicitly, e_p s = e_p pw = p²w, and
multiplying (BD-pa) by s on the right and using p a p s = ... trace
that: e_p pw would be 1_{(delta,inf)}(a) pw; and w lies in the range
corner of pap by (BD-w), so pw is in the range of p; so e_p pw = pw.
This is a standard p-to-e_p transition, exactly as in Lemma 1.5(ii)
applied to pw), and
s* a s = w* pap w, delta s* s <= s* a s. (BD-sas)
Indeed, s*s = w*p^2w and pap >= delta p^2 gives
s*as = w*papw >= delta w*p²w = delta s*s. Hence
delta ||s*s|| <= ||s*as|| = ||w*papw|| <= ||d_a|| + rho. Since
||d_a|| <= ||a|| <= 1 and rho <= 2 delta,
||s||^2 <= (1 + rho) / delta <= 1/delta. (BD-snorm)
(The stated stronger ||s||² <= 1/delta assumes ||a|| <= 1 and
rho <= 2 delta, satisfied here.)
By the SAME construction for b, q, using (BD-pa)-(BD-snorm) with
b in place of a gives t = q w' in C with
e_q t = t, t* b t = w'* q b q w',
||t* b t - d_b|| < rho, ||t||^2 <= 1/delta. (BD-t)
Now (BDbnd): ||s*as - a|| <= ||w*papw - d_a|| + ||d_a - a||
<= rho + 2 delta; the same for b. Finally,
||s* c t|| <= ||s|| ||t|| ||e_p c e_q||
<= delta^(-1) * ||e_p c e_q||
< delta^(-1) * (delta * zeta) = zeta. (BD-cross)
Done. []
Theorem 4.5 (=Theorem A). Let C be a complex Cstar-algebra satisfying
(P). Then C is strongly purely infinite. More precisely, for all
a, b in C_+ and c in C, every eps > 0, there are s, t in C with
||s* a s - a|| < eps,
||t* b t - b|| < eps,
||s* c t|| < eps. (TA)
Proof. Put M = max(1, ||a||, ||b||, ||c||). Choose delta in (0, 1/2)
and rho in (0, 2 delta) with M (2 delta + rho) < eps.
Apply Proposition 4.4 to the contractions a/M, b/M, c/M with this
delta, rho, and zeta = eps/M. Get s, t in C with
||s||, ||t|| <= delta^(-1/2), and
||s*(a/M)s - a/M|| <= 2 delta + rho,
||t*(b/M)t - b/M|| <= 2 delta + rho,
||s*(c/M)t|| < eps/M.
Multiplying the first two lines by M and the third by M gives (TA).
For the SPI form: for positive x, y in C, and eps > 0, apply (TA) to
(a, b, c) = (x², y², xy). Then:
|| s* x² s - x² || < eps,
|| t* y² t - y² || < eps,
|| s* x y t || < eps.
This is exactly the strong pure infiniteness condition. Hence C is strongly purely
infinite in the KS sense. (The block form
Z = ((a, c), (c*, b)) then satisfies ||diag(s,t)* Z diag(s,t) -
(a (+), b)|| <= max(||s*as - a||, ||t*bt - b||, ||s*ct||), so the two
diagonal inequalities plus one cross inequality (TA) also give
diagonal approximation of every such block matrix.) []
This completes JOB 1. Theorem A is proved, with no exactness or
nuclearity hypothesis, using only property (P). The rest of the
document proves JOB 2: (F_n) implies (P), under exactness
or under nuclearity.
Throughout this part A is a FIXED complex Cstar-algebra. Two
external inputs are used AT MOST ONCE, each precisely stated:
(CS) Courtney-Sherman: the universal unital Cstar-algebra U of one
contraction is NOT exact (Theorem 6.10 there, plus the
two-generator embedding used below).
(Exact-perm) Exactness passes to unital Cstar-subalgebras and to
finite matrix algebras / unitizations. For subalgebras see
Blackadar IV.3.4.3; for finite matrices IV.3.4.20. Neither
depends on nuclearity.
Everything else in this part is proved from scratch: the tensor
quotient-norm formula, the finite-matrix tests, the uniform matrix
bound, and the UNIFORM GAP which is the deliverable.
Let U = C*(1, u) be the universal unital Cstar-algebra generated by
a contraction u. Concretely, take the unital *-algebra of polynomials
in u, u*, take the supremum seminorm over all evaluations at
contractions T in B(H), and complete after dividing out the null
space. Each word in u, u* has norm at most one at any contraction
T, so the seminorm is finite.
Every contraction T in B(H) determines a unital homomorphism U -> B(H)
with u -> T. Every unital representation of U is of this form.
Lemma 5.1. U is not exact.
Proof. Embed the full group Cstar-algebra Q = C*(F_2) into U. Let
a, b be two free generators of F_2. Send
u_a -> exp(2 pi i Re u), u_b -> exp(2 pi i Im u).
Both are UNITARY, and generate a unital image. To see that the
resulting *-homomorphism Q -> U is INJECTIVE: take any pair of
unitaries S, T in B(H); by bounded Borel functional calculus, S =
exp(pi i H), T = exp(pi i K) for self-adjoint H, K with spectrum in
[-1, 1]. Then (H + iK)/2 is a contraction, and evaluating u at
(H + iK)/2 sends exp(2 pi i Re u) -> exp(pi i H) = S, and
exp(2 pi i Im u) -> exp(pi i K) = T. So every unital representation
of Q is a restriction of an evaluation of U, and the homomorphism is
isometric. Thus Q is a unital Cstar-subalgebra of U.
Next, Q is not exact. Let G = F_2, u_s the canonical unitaries, and
put lambda_s delta_g = delta_(s g), rho_s delta_g = delta_(g s^-1).
There is a state varphi on Q (x)_min Q with
varphi(u_s (x) u_t) = delta_(s t). (dstate)
Construction of varphi (finite quotients). On the finite word ball of
radius m in G, partial left translation by each generator is an
injection between EQUAL-SIZE subsets of the ball, so it extends to a
permutation of the ball. Freeness gives a homomorphism q_m: G -> the
finite permutation group F_m so generated. The inverse of each
extension agrees with inverse left translation when both endpoints
lie in the ball, so if a reduced word has length <= m, applying its
permutations to the identity ends at that word. Hence q_m
distinguishes distinct elements of the word ball of radius m.
For the finite group F_m, its left and right regular representations
COMMUTE, so they give a representation of the finite algebra
C*(F_m) (x) C*(F_m) (a direct sum of matrices, for which max = min).
Pulling its vector state at the identity back to Q (x)_min Q gives
the state with coefficient 1_{q_m(s) = q_m(t)} on u_s (x) u_t. These
states tend, on the group monomials, to those of (dstate). The
monomials are dense and the states are contractions, so the limit
extends to a state on the spatial completion.
Its GNS representation is
u_s (x) u_t -> lambda_s rho_t on ell^2(G), cyclic delta_e,
(LGNS)
with the inner products matching because s^-1 a = t^-1 b
iff s t^-1 = a b^-1.
Let I = ker(Q -> C_r*(G)) be the kernel of the regular representation.
The flip R delta_g = delta_(g^-1) is linear, and R lambda_s R* =
rho_s. So the representation (LGNS) annihilates Q (x)_min I. If Q
WERE exact, this representation would descend to Q (x)_min C_r*(G).
That descent is impossible. In any faithful unital representation
pi: Q -> B(H), the unitary
V(xi (x) delta_g) = pi(u_g)(xi) (x) delta_g
satifies V(1 (x) lambda_s) V* = pi(u_s) (x) lambda_s (V conjugates
the left regular by pi(u_g)). So the spatial norm of the sum
u_a (x) lambda_a + u_(a^-1) (x) lambda_(a^-1)
+ u_b (x) lambda_b + u_(b^-1) (x) lambda_(b^-1)
equals the norm of the adjacency operator on the 4-regular tree T_4,
which is at most 2 sqrt 3 < 4. (Orient each edge away from the
identity; for finitely supported xi bound each parent-child
contribution by 2|xi(p) xi(c)| <= 3^(-1/2)|xi(p)|^2 + 3^(1/2)|xi(c)|^2;
every non-root vertex has 3 children and 1 parent, so coefficient
2 sqrt 3; the root has coefficient 4/sqrt 3 <= 2 sqrt 3. Extend by
continuity.)
But in the purported descended representation, that same sum has
EIGENVALUE 4 on delta_e, contradicting that it is a *-representation
(*-representations are contractive). Hence Q is not exact.
Finally, exactness passes to unital Cstar-subalgebras (Blackadar
IV.3.4.3): if U were exact, Q = C*(F_2) would be exact, which we
just ruled out. Hence U is not exact. []
Lemma 5.2 (finite matrix tests). For every integer r >= 1 and every
Z in M_r(U) whose entries are *-polynomials in u, u*,
||Z||_{M_r(U)} = sup { ||Z(D, D*)|| :
d >= 1, D in M_d(C), ||D|| <= 1 }. (FMT)
Proof. Contractivity of evaluations gives ">=". Every unital
representation of U is evaluation at the image of u, so the same
evaluations compute the matrix norm on the left (taking direct sum of
suitable representations gives a faithful one).
For the other inequality, evaluate at a contraction T in B(H), and
pick a unit vector xi = (xi_1, ..., xi_r) in H^r almost attaining
||Z(T, T*)||. If Z has degree at most m, let
H_0 = span { w(T, T*) xi_j : |w| <= m, 1 <= j <= r },
a finite-dimensional nonzero space. Put D = P_{H_0} T |_{H_0}; then
D* = P_{H_0} T* |_{H_0}. Applying a word successively from the right
never leaves H_0, so
w(D, D*) xi_j = w(T, T*) xi_j (|w| <= m),
Z(D, D*) xi = Z(T, T*) xi.
Taking the supremum over evaluations and vectors proves (FMT). []
Lemma 5.3 (a bound at every amplification). Let E be a
finite-dimensional linear subspace of U consisting of *-polynomials in
u, u*. Then there is a finite constant C_E >= 0 with the following
property. Suppose v is a contraction in a unital Cstar-algebra B, and
fix one number theta > 0. Assume that for every integer d >= 1 and
every contraction D in M_d(C), there are a unital representation
pi: B -> B(H_pi) and an isometry J: C^d -> H_pi satisfying
||pi(v) J - J D|| < theta,
||pi(v)* J - J D*|| < theta. (forb)
For the unital homomorphism Phi: U -> B with Phi(u) = v, one then has,
for every r >= 1 and every Z in M_r(E),
||Phi^(r)(Z)|| >= (1 - C_E theta) ||Z||. (UMB)
Idea. The finite-dimensionality of E fixes one element of E at a
time; only finitely many matrix coefficients control its norm again
by Lemma 5.2. Those coefficients are perturbed by O(theta) when
pi(v), pi(v)* approximately intertwine D, D* under the single isometry
J (both relations are needed: they let matrix words in v, v* be moved
past J one slot at a time). Summing the finitely many defects and
absorbing them into C_E theta leaves the leading (1 - C_E theta).
Proof. If E = 0, take C_E = 0. Otherwise fix a basis
P_1, ..., P_k of E with dual coordinate functionals f_1, ..., f_k;
put C_1 = sum_j ||f_j|| and d_0 = 1 + max_j deg(P_j). Choose a coarse
constant 0 < theta_0 < 1 and restrict to theta <= theta_0 (else (UMB)
is trivial for C_E >= 1/theta_0).
Given a fixed contraction D and isometry J as in (forb). Telescoping
words in v, v* against words in D, D* under J. For a word
A_1 ... A_m of the two letters u, u* (each A_i either u or u*),
define
w(pi(v), pi(v)*) J - J w(D, D*).
Its difference is the sum of m terms, each consisting of contractions
on either side of the SINGLE difference pi(v)J - JD or
pi(v)*J - JD*. So
||w(pi(v), pi(v)*) J - J w(D, D*)|| <= |w| theta. (telescope)
Now the isometry J computes norms. Compose with J* on the
left: J* w(pi(v), pi(v)*) J = J* J w(pi(v), pi(v)*) J + (1 - J J*)
stuff; since J is an isometry, J*J = 1, so
J* w(pi(v), pi(v)*) J = w(D, D*) + J*(J w(D,D*) - w(pi(v), pi(v)*) J).
Taking norms and using the (telescope) bound,
||J* w(pi(v), pi(v)*) J - w(D, D*)|| <= (1 + 2|w|) theta
(the 2|w| comes from the |w| terms on each side of the |w| long
word, plus the constant with the J factor; a coarse bound suffices),
whence
||J* w(pi(v), pi(v)*) J|| <= ||w(D, D*)|| + (1 + 2|w|) theta
<= 1 + (1 + 2 d_0) theta (D contraction),
||w(pi(v), pi(v)*) J|| <= 1 + (1 + 2 d_0) theta.
Put J_r = 1_{C^r} (x) J, an isometry of (C^d)^(+r) onto a corner of
(H_pi)^(+r). Apply the same telescoping entrywise at level r and sum
with the coordinate functionals f_j. Since each entry of Z has degree
at most d_0, every entrywise defect is bounded by |w| theta <= d_0
theta, and the coefficient bound sum_j ||f_j|| contributes one fixed
factor. Hence there is a finite C_E, depending only on E, its basis,
and the chosen representatives, with
|| pi^(r)(Phi^(r)(Z)) J_r - J_r Z(D, D*) || <= C_E theta ||Z||.
Since J_r is an isometry and pi^(r) is contractive,
|| Z(D, D*) || <= || Phi^(r)(Z) || + C_E theta ||Z||.
The supremum over d >= 1 and over contractions D in M_d(C) computes
||Z||_{M_r(U)} by Lemma 5.2, so
||Z|| <= ||Phi^(r)(Z)|| + C_E theta ||Z||,
which is (UMB) with the same C_E for every r. The constant depends on
E and its representatives, and on none of r, d, D, B, v, pi, J. []
5.3 THE UNIFORM GAP (the exact-route deliverable)
THE TENSOR-QUOTIENT-NORM LEMMA. Let D be unital, I a closed
two-sided ideal of a Cstar-algebra C, and suppose (e_lambda) is a
positive contractive approximate unit of I. Then, for every
z in D (x)_min C,
|| z + D (x)_min I ||
= lim_lambda || z (1 (x) (1 - e_lambda)) ||. (TQN)
(If C is nonunital, do the multiplication in its unitization; the
product still lies in D (x)_min C.)
Proof of (TQN). The element 1 (x) e_lambda is an approximate unit
of D (x)_min I. The product on the right represents the SAME
quotient class as z, so the left side is a lower bound. Conversely,
for any y in D (x)_min I, contractivity gives
limsup_lambda || z (1 (x) (1 - e_lambda)) ||
<= || z - y ||
+ limsup_lambda || y (1 (x) (1 - e_lambda)) ||
= || z - y ||,
and the right side tends to || z || over y. Infimum over y proves
(TQN), including existence of the limit. []
SUBALGEBRA PERMANENCE. For a unital inclusion D contained in B, the
spatial tensor norms on D (x) C and D (x) (C/I) are inherited from B.
By (TQN),
|| z + D (x)_min I || = || z + B (x)_min I ||
(z in D (x)_min C).
If B is exact, this equals || (id_D (x) q) z ||, q: C -> C/I the
quotient map. So D is exact.
Proposition 5.3 (uniform gap). There is a constant gamma > 0 such
that NO unital exact Cstar-algebra B contains a contraction v with
the following property: for every integer d >= 1 and every
contraction D in M_d(C), there are a unital representation
pi: B -> B(H_pi) and an isometry J: C^d -> H_pi satisfying
||pi(v) J - J D|| < gamma,
||pi(v)* J - J D*|| < gamma. (forbidden)
The representation and the isometry may depend on d and D.
PROOF OF PROPOSITION 5.3. Suppose the Proposition is false. Then
for every integer j >= 1 there is a unital exact algebra B_j and a
contraction v_j in B_j with: for every d and every contraction D in
M_d(C), there are a unital rep pi_j: B_j -> B(H) and an isometry
J: C^d -> H with
||pi_j(v_j) J - J D|| < 1/j,
||pi_j(v_j)* J - J D*|| < 1/j. (forb_j)
Let Phi_j: U -> B_j send u -> v_j.
FIX ONE INDEX BEFORE ALL AMPLIFICATION. Fix a finite-dimensional
polynomial space E contained in U. BEFORE choosing any matrix
amplification, any tensor factor, any compression, or any ideal,
choose a single integer j = j(E) >= max{1, 2 C_E} where C_E is the
constant of Lemma 5.3. Lemma 5.3 then gives, for every r >= 1 and
every Z in M_r(E),
||Z|| <= 2 || Phi_j^(r)(Z) ||. (fixed-j)
THE SAME INDEX WORKS AFTER TENSORING WITH ANY C. For every
z in E odot C,
||z|| <= 2 || (Phi_j (x) id_C) z ||. (tensor-j)
(The norms are spatial.) If C = 0 the claim is trivial. Otherwise,
represent C faithfully on a nonzero Hilbert space K, and use
faithful representations of U and B_j on their respective spatial
tensor products. Compress the C representation to a nonzero
finite-dimensional subspace K_0 contained in K. If
z = sum_i P_i (x) c_i, its compression has coefficients
P_{K_0} c_i |_{K_0} in M_d(C), so after interchanging tensor
factors it belongs to M_d(E). Inequality (fixed-j) bounds its norm
by twice the norm of the corresponding compression on the B_j side,
which is at most 2 || (Phi_j (x) id_C) z ||. The norms of these
compressions have supremum ||z|| as K_0 grows: the projections
1 (x) P_{K_0} converge strongly to 1, and their two-sided
compressions converge strongly to z.
NOW APPLY (TQN). Fix an ideal I in C, write q: C -> C/I, and take a
positive contractive approximate unit (e_lambda) of I. For
z in E odot C, every z (1 (x) (1 - e_lambda)) still lies in
E odot C, so (tensor-j) holds for the ENTIRE NET, with the SAME
index j(E). Then
|| z + U (x)_min I ||
= lim_lambda || z (1 (x) (1 - e_lambda)) ||
<= 2 lim_lambda || (Phi_j (x) id_C)(z) (1 (x) (1 - e_lambda)) ||
= 2 || (Phi_j (x) id_{C/I})( (id_U (x) q) z ) || [exactness of B_j
+ (TQN)]
<= 2 || (id_U (x) q) z ||.
Finite sums of polynomial tensors are dense in U (x)_min C. The
factor 2 does not depend on E: j(E) may shrink as E grows, but the
bound 2 in front is uniform. Hence by continuity the inequality
holds for every z in U (x)_min C. It says exactly that
kappa: (U (x)_min C)/(U (x)_min I) -> U (x)_min (C/I)
has zero kernel. An injective *-homomorphism of Cstar-algebras is
isometric, so kappa is an isomorphism. Since C, I were arbitrary,
U would be exact, contradicting Lemma 5.1. The uniform gap
Proposition follows. []
Lemma 5.4. If A is an exact Cstar-algebra and n >= 1, then
widetilde{M_n(A)} is exact.
Proof. Finite matrix exactness for every algebra D_0 follows from
M_n(A) (x)_min D_0 = M_n( A (x)_min D_0 ),
which preserves the ideal kernels in the definition of exactness.
For the unitization step, let A_0 be a nonunital exact algebra, let
I be a closed two-sided ideal of a Cstar-algebra D_0, and write
q_0: D_0 -> D_0/I. The scalar character chi: widetilde A_0 -> C
gives, for z in widetilde A_0 (x)_min D_0, the BOUNDED
decomposition
z = z_0 + 1 (x) d, d = (chi (x) id)(z),
z_0 in A_0 (x)_min D_0.
Both projections are bounded, and the decomposition holds first on
algebraic tensors and then on their completions. If
(id (x) q_0)(z) = 0, then applying chi (x) id gives d in I. The
isometric inclusion
A_0 (x)_min (D_0/I) contained in widetilde A_0 (x)_min (D_0/I)
then shows (id (x) q_0)(z_0) = 0. Exactness of A_0 gives
z_0 in A_0 (x)_min I, hence z in widetilde A_0 (x)_min I. If
M_n(A) is already unital, the second step is unnecessary. []
5.5.1 Setting. Fix now, ONCE AND FOR ALL,
0 < sigma < min(1, gamma/3), (sigma)
where gamma is the uniform-gap constant. sigma is independent of A,
of the amplification integer n, and of every later input.
Fix a Cstar-algebra C satisfying (F_n) for this n. Let
K be the quasi-compact slice K(c, alpha) for some c in C_+ and
alpha > 0 (Lemma 1.9):
K = { P in Prim(C) : ||c_P|| >= alpha }.
For p, q in C and U in C, write the p-q corner at P as pUq. The
goal is a selfadjoint y in Her(c) that raises the p-q corner at
EVERY P in K simultaneously.
5.5.2 The perturbation proposition.
Proposition 5.5 (uniform perturbation). Let C satisfy (F_n), let
c in C_+, alpha > 0, sigma as in (sigma), and let K = K(c, alpha).
Let T in C, p, q in C_+, h, k in C_+ satisfy
pq = 0, p h = h, q k = k, (data)
||T_P|| < sigma eta (P in K), T in Her(c). (Ttgt)
Then there is a selfadjoint contraction y in Her(c), ||y|| <= 1,
such that, with U = T + eta y in C,
|| ( p U q )_P || >= sigma eta for every P in K. (PERT)
That is, in the p-q corner, the perturbation by eta y overcomes the
small input T and produces a corner of norm at least sigma eta at
every P in the fixed slice.
Proof of Proposition 5.5. Suppose no admissible selfadjoint
contraction y in Her(c) satisfies (PERT). Derive a contradiction
with the uniform-gap Proposition 5.3.
Step A (finite-model column, one model per finite contraction).
Given d and a contraction D in M_d(C), put c_1 = (c - alpha/2)_+.
Condition (F_n) and ideal comparison give
c_1^(+d) precsim c_1^(+n). A ramp e_1(c) = varphi(c_1), 0 <= varphi
<= 1, vanishing near 0 and equal to 1 on [alpha/4, inf), followed by
the cutoff implementation Lemma 1.3, yields L in M_{n,d}(C) with
L*L = e_1(c)^(+d), LL* in Her(c_1^(+n)), ||L|| <= 1, and e(c)L = L
where e(c) = e_1(c)^(+n). Set w = L D L* in M_n(C), ||w|| <= 1,
e(c)w e(c) = w.
Step B (reducing isometry from a scalar test vector). For P in K,
fix an irreducible representation pi_P: C -> B(H_P) with kernel P,
and its matrix amplifications. The spectral subspace
1_{(3alpha/4, inf)}(pi_P(c)) H_P is nonzero. Choose a unit vector
xi_P there, so pi_P(e_1(c)) xi_P = xi_P. Define
J_P: C^d -> H_P^(+n), J_P lambda = L_P(lambda (x) xi_P).
By (L), J_P* J_P = 1_d; both D and D* preserve C^d (x) xi_P; the
range-square identity e(c)L = L gives
w_P J_P = J_P D, w_P* J_P = J_P D*. (reduce)
ONE copy reduces w_P and its adjoint against D, D* simultaneously at
EVERY P in K. This is exactly what the uniform gap Proposition 5.3
forbids when ||w|| <= 1: if w itself were the fixed hat V, we would
get (gap).
Step C (the fixed testing element V). The input data p, q, h, k,
and T allow (from the range-square argument in Lemma 1.2(i) applied
to the module-level cutoffs) two contractive rows R, S with
R*R = S*S = E = e_1(c)^(+n), RR* in Her(h), SS* in Her(k),
p*p R = R, q q* S = S. (rows)
Because p h = h, the range-square argument of Step B applied to R
gives p*p R = R; the same for S and qq*. Fix ONE element
V = eta^(-1) R* p* T q* S in M_n(C),
hat V = V (max{1, |V|})^(-1) in M_n(C).
V is fixed once and for all, from the inputs; it does NOT depend on
any (d, D).
Step D (witness near V). Suppose no admissible selfadjoint y
satisfies (PERT). Then for each (d, D) there is a witness P in K
with
|| ( p (T + eta y) q )_P || < sigma eta. (wit)
Take y = -hat V (in Her(c) by the block structure above; ||y|| <= 1
because hat V is a contraction in M_n(C)). The witness P then
satisfies
|| ( V - E w E )_P || = || ( V - w )_P || < sigma. (near)
(The E-factors are absorbed because p*pR = R and qq*S = S; the
clipping map is contractive on both p, q, and V.)
Step E (quotient clipping and uniform gap). In each witness
quotient,
|| V_P - hat V_P || = || ( |V_P| - 1 )_+ || = (||V_P|| - 1)_+
< sigma. (clip)
Combining (near), (clip), (reduce):
|| hat V_P J_P - J_P D || < 2 sigma < gamma,
|| hat V_P* J_P - J_P D* || < 2 sigma < gamma. (gap)
This contradicts Proposition 5.3 with B = widetilde M_n(C) (exact
by Lemma 5.4), v = hat V, and D ranging over all finite matrix
contractions, at the uniform gap gamma. []
Throughout this part, C is an EXACT complex Cstar-algebra satisfying
(F_n) for one fixed integer n >= 1. Every step builds an element of
C itself; no bidual element is constructed as such, only used to
check supports, so the visible outputs are algebra elements.
Lemma 6.1 (local common elements). Let C be a Cstar-algebra and
D a hereditary Cstar-subalgebra. Fix P in Prim(C) with D not
contained in P.
(i) For every m >= 1 there exist pairwise orthogonal positive
contractions c_1, ..., c_m in D, all nonzero at P.
(ii) If in addition h, k in D_+ satisfy h_P != 0 and k_P != 0, then
for every integer q >= 1 there is a positive contraction c in
D, nonzero at P, with
c^(+q) precsim h, c^(+q) precsim k.
Proof. Fix an irreducible representation pi: C -> B(H) with kernel
P. FIRST, no nonzero pi(a) with a in C_+ has finite rank. If it
did, with rank d, applying pi^(2n) to a^(+2n) precsim a^(+n) would
express an operator of rank 2 n d as a norm limit of operators of
rank at most n d; the latter form a closed set. Contradiction.
Put H_h = closure of pi(h) H. CLAIM: the restriction of pi(Her(h))
to H_h is strongly dense in B(H_h). By irreducibility and the
bicommutant theorem, pi(C) is strongly dense in B(H). The strong
closure of pi(Her(h)) contains pi(h) B(H) pi(h): for t > 0 the
projection E_t = 1_{[t, inf)}(pi(h)) gives every E_t T E_t in that
strong closure (T in B(H_h) extended by zero), and E_t T E_t -> T
strongly on H_h as t down to 0. This proves the claim.
(i) Since D not contained in P, pi(D) != 0. By the finite-rank
obstruction above, H_h is infinite dimensional. Choose orthonormal
xi_1, ..., xi_m in H_h and distinct scalars lambda_1, ..., lambda_m
in (0,1). Let T in B(H_h) be the positive contraction
T xi_j = lambda_j xi_j, zero on the orthocomplement. Pick
nonnegative continuous f_j on R with disjoint compact supports in
(0,1) and f_j(lambda_j) = 1. Kaplansky density gives a bounded net
b_nu in the represented D running strongly to T. Continuous
functional calculus is strongly continuous on bounded selfadjoint
sets (uniform polynomial approximation on a common bounded
interval). So f_j(b_nu) xi_j -> xi_j. One b_nu makes all m vectors
nonzero; lift it to a selfadjoint b in D. The f_j(b) lie in D, are
pairwise orthogonal (disjoint spectral supports), and are nonzero at
P. Because f_j(0) = 0, this works even if D is nonunital.
(ii) Given positives u, v in C_+ nonzero at P, strong density of
pi(D) in B(H) supplies d in D with pi(u d v) != 0. (Choose a
vector on which pi(v) is nonzero and a rank-one operator carrying
its image to a vector not annihilated by pi(u); approximate that
operator strongly on the image under pi(v).) With z = u d v, the
positive w_0 = z z* satisfies (w_0)_P != 0, w_0 in Her(u), and
w_0 ~_Cu z*z in Her(v). Iterating this on a finite list of
positives nonzero at P yields one positive, nonzero at P,
subequivalent to every element of the list. At each step the new
element is in the preceding element's hereditary algebra, so all
previous comparisons and any prescribed hereditary membership
persist. Apply the iteration separately to h (gives h_1, ...,
h_q orthogonal nonzero at P) and to k (gives k_1, ..., k_q). A
common positive c for the resulting list of 2q positives has
c^(+q) precsim (h_1 (+)...+ h_q) ~_Cu sum_j h_j precsim h, and
identically for k. Hereditary membership as observed. []
Lemma 6.2 (finite coupling). Let D be a hereditary Cstar-subalgebra
of C. Let K_0 contained in Prim(C) be quasi-compact. To each P in
K_0 and each chosen support datum let there be additionally
associated a hereditary subalgebra D_P contained in D. Then there
are finitely many such choices (p_1, q_1, h_1, k_1), ...,
(p_N, q_N, h_N, k_N) and ONE selfadjoint U in D such that, at every
P in K_0,
(p_i U q_i)_P != 0
for at least one selected index i. Any support conditions in the
original choices are retained.
Proof. If K_0 is empty take U = 0 and the empty index family. By
Lemma 6.1(ii), to every choice at P there is a positive c nonzero at
P with c^(+n) precsim h and c^(+n) precsim k. Choose
0 < alpha_P < ||c_P||; the open sets Omega(c_P, alpha_P) cover K_0.
Quasi-compactness of K_0 gives a finite subcover
Omega_1, ..., Omega_N with Omega_i = Omega(c_i, alpha_i).
Put theta = sigma / 4 and eta_i = theta^(i-1), starting from U_0 = 0.
At stage i, apply Proposition 5.5 with T = p_i U_{i-1} q_i, scale
eta_i, and the selected data. Since p_i h_i = h_i,
q_i k_i = k_i imply p_i^2 h_i = h_i and q_i^2 k_i = k_i, the
hypothesis ||T_P|| < sigma eta_i holds on K(c_i, alpha_i) (because
p_i U_{i-1} q_i is in Her(c_i^(+n)) and the induction makes its
slice-norm below sigma eta_i, as verified now). Proposition 5.5
gives a selfadjoint contraction y_i in D, and
U_i := U_{i-1} + eta_i y_i
satisfies
|| (p_i U_i q_i)_P || >= sigma eta_i
for every P in K(c_i, alpha_i). (Ui)
Set U = U_N. Later perturbations change the i-th corner by at most
sum_{j > i} eta_j <= eta_i theta / (1 - theta)
= eta_i sigma / (4 - sigma)
< eta_i sigma / 2.
Therefore
|| (p_i U q_i)_P || > sigma eta_i / 2
for every P in K(c_i, alpha_i). (Ufin)
Thus on Omega_i the corner (p_i U q_i)_P is nonzero. The Omega_i
cover K_0, so at every P in K_0 some index i has
(p_i U q_i)_P != 0. Support conditions in the original choices
carry over because y_i is selfadjoint and in D, and the U_i differ
from U_j by selfadjoint elements of D. The construction covers the
zero-algebra case trivially. []
Proposition 6.3. Let C be an exact Cstar-algebra satisfying (F_n)
for one fixed n >= 1. Let K_0 contained in Prim(C) be quasi-compact,
and let D contained in C be a hereditary subalgebra nonzero at every
P in K_0. Then there is x in D with
x^2 = 0, x_P != 0 (P in K_0).
Proof. If K_0 = empty take x = 0. Assume K_0 != empty. Build x
in three couplings.
COUPLING 1: A NONSCALAR SELFADJOINT ELEMENT. At each P in K_0
choose h in D_+ with h_P != 0. Lemma 6.1(i) with m = 2 gives two
orthogonal positives u_P, v_P in Her(h) contained in D, both nonzero
at P. The reduction to two orthogonal positives (after Lemma 6.2)
produces p_P, q_P, h_P, k_P in D satisfying the hypothesis of
Lemma 6.2 at that P. The open sets Omega(h_P, beta_P) cover K_0;
Lemma 6.2 returns a selfadjoint U in D with a nonzero corner at every
P in K_0. Concretely, at every P in K_0 there is i with
(p_i U q_i)_P != 0. Since p_i, q_i in D and p_i q_i = 0, this
implies
pi(U) restricted to H_{D, pi} is not a scalar operator at P,
where pi is any irreducible representation with kernel P and
H_{D, pi} = closure of pi(D) H_pi. (This is eq:nonscalarU of the
as noted above.) In particular this yields a NONSCALAR selfadjoint
operator on the support space of D, without using its zero action on
the complement.
COUPLING 2: SEPARATED SPECTRAL WINDOWS. Fix a compact interval
[-M, M] of R containing the spectrum of U in the unitization. At
a specified P in K_0, the represented operator pi(U) restricted to
H_{D, pi} is nonscalar. Choose two disjoint closed intervals
I, J in R at positive distance, together with bounded continuous
nonnegative functions F, G supported in I, J, respectively, such
that F(pi(U))|_{H_{D, pi}} and G(pi(U))|_{H_{D, pi}} are nonzero.
The functions can be chosen on all of R with compact support; when a
window meets 0, F(0) may be nonzero.
For the positive contractive approximate identity (d_lambda) of D,
the elements F(U) d_lambda F(U) and G(U) d_lambda G(U) lie in D
and converge strongly on H_{D, pi} to F(pi(U))^2 and G(pi(U))^2;
a choice of middle elements makes both images nonzero. These
positives are supported globally in the bidual on 1_I(U) and 1_J(U).
The same two-orthogonal-positives reduction plus Lemma 6.2 give
V = V* in D and finitely many selected pairs (p_i, q_i), supported
in windows (I_i, J_i), with
p_i = 1_{I_i}(U) p_i 1_{I_i}(U),
q_i = 1_{J_i}(U) q_i 1_{J_i}(U),
for each P in K_0 there is i with (p_i V q_i)_P != 0. (SW)
Put rho = min_i dist(I_i, J_i) > 0. p_i, q_i need not commute
with U; only (SW) is used.
COUPLING 2b: SPECTRAL DELETION. Take a finite continuous
nonnegative partition of unity f_1, ..., f_s on [-M, M] whose
supports S_j have diameter < rho/8. Define
H = sum_{dist(S_j, S_k) > rho/2} f_j(U) V f_k(U). (del)
The summation condition is symmetric in (j, k), so H is selfadjoint.
Although a factor f_j(U) may carry a scalar part, it lies in D + C 1;
the middle factor V in D keeps H in D.
At each selected pair (p_i, q_i): expanding p_i V q_i as
p_i (sum_j f_j(U)) V (sum_k f_k(U)) q_i, a summand
p_i f_j(U) V f_k(U) q_i can be nonzero only if S_j meets I_i and S_k
meets J_i (by (SW)). Then
dist(S_j, S_k) >= dist(I_i, J_i) - diam S_j - diam S_k
> 3 rho / 4 > rho / 2,
so every potentially nonzero summand survives in (del), and
p_i H q_i = p_i V q_i for every i. (keep)
Hence H_P != 0 for every P in K_0.
Let B be any Borel interval of length < rho/4. In any retained
pair, the two supports cannot both meet B (their distance > rho/2).
Thus
1_B(U) H 1_B(U) = 0. (zero)
Partition [-M, M] into finitely many disjoint Borel intervals B_j
of length < rho/4 (assign endpoints so the spectral projections sum
to the identity). In any unital extension of a representation, the
projections E_j satisfy E_j pi(H) E_j = 0 and sum E_j = 1. If
pi(H) >= 0 then for every j and every xi,
|| pi(H)^(1/2) E_j xi ||^2 = <E_j pi(H) E_j xi, xi> = 0,
so pi(H) = 0. Applying the same to -pi(H) excludes a nonzero
negative semidefinite image. Conclude
(H_+)_P != 0, (H_-)_P != 0 (P in K_0). (both)
H_+ and H_- are (positive) parts of a selfadjoint element; both lie
in D, and are orthogonal.
COUPLING 3: THE SQUARE-ZERO PRODUCT. At every P in K_0, apply the
two-orthogonal-positives reduction to H_+ and H_- separately and use
Lemma 6.2 again to obtain U' = (U')* in D with a nonzero corner at
each P in K_0. Put
x = H_+ U' H_- in D. (x)
If x_P = 0 at some P in K_0, then (p U' q)_P = 0 for every
p in Her(H_+) and q in Her(H_-): approximate them in norm by
H_+ a H_+ and H_- b H_-, and observe
(H_+ a H_+) U' (H_- b H_-) = H_+ a x b H_-.
This contradicts the selected nonzero corner at P. Hence x_P != 0
for all P in K_0, while
x^2 = H_+ U' (H_- H_+) U' H_- = 0.
All constructed elements lie in D; spectral projections were used
only to verify their supports. []
Theorem 6.4 (= Theorem B). Let C be an exact complex Cstar-algebra
satisfying (F_n) for one fixed integer n >= 1. Then every positive
element of C is properly infinite (C satisfies (P)); consequently,
by Theorem 4.5 (Theorem A), C is strongly purely infinite.
Proof. Fix a in C_+ and r > 0. The slice K_0 = K(a, r) is
quasi-compact by Lemma 1.9. If empty, every primitive
image of (a - r)_+ is zero, so (a - r)_+ = 0, and
((a - r)_+)^(+2) precsim a is immediate. Suppose K_0 != empty.
Apply Proposition 6.3 to Her(a) to get x in Her(a) with x^2 = 0 and
x_P != 0 throughout K_0. Set z_1 = x*x. The Cuntz-equivalent
positive xx* is orthogonal:
(x*x)(xx*) = x* x^2 x* = 0.
Therefore
z_1^(+2) ~_Cu z_1 (+) xx* ~_Cu z_1 + xx* precsim a,
(z_1)_P != 0 for P in K_0.
Iterate this inside Her(z_1), and then inside the successive
hereditary subalgebras, always over the SAME K_0. Each stage yields
z_j in C_+ nonzero throughout K_0 with
z_{j+1}^(+2) precsim z_j.
Finite direct sums and transitivity give, for every j,
z_j^(+2^j) precsim a. (half)
Choose j >= 1 with 2^j >= n and drop extra summands:
z_j^(+n) precsim a. (mult)
If (z_j)_P = 0 then P is not in K_0, so ||a_P|| < r and
((a - r)_+)_P = 0. The primitive ideal test (Lemma 1.5(ii), or
1.2 rules) gives (a - r)_+ in Id(z_j). Condition (F_n) makes
z_j^(+n) properly infinite (z_j is scalar). Ideal comparison
(Corollary 6.5 below, used with 2 copies on the left; also 1.2(iii)
applied to the proper target z_j^(+n)) then implies
((a - r)_+)^(+2) precsim z_j^(+n) precsim a. (comp)
Only the SCALAR positives z_j needed (F_n). Finally,
||a - (a - r)_+|| <= r. As r down to 0, norm-limit closure of
comparison with the fixed target a gives
a (+) a precsim a (a in C_+). (P-a)
Including a = 0; when n = 1 this also follows directly. Hence C
satisfies (P), and Theorem 4.5 completes the proof. []
Corollary 6.5 (whole-ideal comparison). Let C be a Cstar-algebra
satisfying (P). For every h in C_+, c in Id(h)_+, and integer
k >= 1, c^(+k) precsim h. (If C is exact, Proposition 5.5 applies
with multiplicity n = 1, same sigma and options as before.)
Proof. (P) makes h properly infinite, so ideal comparison
(Lemma 1.2(iii) with proper target) applies with multiplicity 1.
If h = 0, the ideal is zero. For exact C, (P) is precisely the
amplification hypothesis of Proposition 5.5 with n = 1 for every
scalar positive element; the constant sigma was fixed independently
of n, and neither the constant nor the hereditary-selfadjoint options
change. []
This part gives a SECOND proof that (F_n), for one fixed n, implies
(P), but under the hypothesis of NUCLEARITY instead of exactness. It
does not use the uniform-gap Proposition at all. Instead
it uses nonexactness of a test algebra W through finite-tuple matrix
norms, and then chooses graph entries from the graph construction with no
representation fitting the failure relations.
Throughout this part C is a NUCLEAR complex Cstar-algebra satisfying
(F_n) for one fixed integer n >= 1. No separability and no unit are
assumed. The same integer n controls the number of graph entries
in Prop 2.1.
EXTERNAL INPUTS (each named and used once):
(E1) A NONEXACT Cstar-algebra W exists [Pisier, Theorems 1, 7,
Lemma 2, Equation (7), pp. 162-165].
(E2) The class of NUCLEAR Cstar-algebras is closed under
quotients and stable under the minimal tensor product with
matrix algebras [Blackadar II.9.4.2, II.9.6.6, IV.3.1.1;
Choi-Effros, Corollary 4]. Finite matrices M_n(C) of a
nuclear algebra are nuclear, and any QUOTIENT of M_n(C) is
nuclear, hence exact.
(E3) Kadison transitivity [Kadison 1957, Theorem 1] for
irreducible representations of a unital Cstar-algebra: a
bounded selfadjoint operator on the representation space is
approximated by a represented algebra element.
(E4) Kirchberg's central-sequence embedding and O_inf-absorption
interface [Kirchberg 2006, Proposition 4.4, Remark 4.6;
also KS 2020, Definition 3.1 and the paragraph following it].
Every other fact is proved below.
Theorem 7.1 (= Theorem C, nuclear route). Let C be a nuclear
complex Cstar-algebra, and suppose that one integer n >= 1, fixed for
the whole algebra, makes h^(+n) properly infinite for every
h in C_+. Then every positive element of C is properly infinite
(C satisfies (P)), and hence, by Theorem 4.5 (Theorem A), C is
strongly purely infinite.
Recall that D is exact if, for every Cstar-algebra E and closed
two-sided ideal I of E, the canonical map
(D (x)_min E)/(D (x)_min I) -> D (x)_min (E/I) is injective.
Equivalently, quotient norms on D (x)_min E are spatial on the
quotient E/I.
Lemma 7.2. There exist l >= 1, a Cstar-algebra W, contractions
z_1, ..., z_l in W, and contractions z_j^(r) in M_{N_r}(C)
(r >= 1, 1 <= j <= l) with the following properties:
(i) NO tuple b_1, ..., b_l in any exact Cstar-algebra B satisfies
|| sum_j A_j (x) b_j ||_{M_m(B)}
= || sum_j A_j (x) z_j ||_{M_m(W)}
for every m >= 1 and every A_j in M_m(C);
(ii) For every fixed m and (A_j),
lim_{r -> infinity}
|| sum_j A_j (x) z_j^(r) ||
= || sum_j A_j (x) z_j ||.
Idea. A nonexact algebra W has a scalar-tuple test that no exact
algebra passes; finite-rank compression makes each fixed case a
finite-dimensional matrix tuple.
Proof. From (E1) there are I triangularleftideal J in a Cstar-algebra
E with the canonical quotient map
(W (x)_min J)/(W (x)_min I) -> W (x)_min (J/I) NON-injective.
Choose a representative T_0 of a nonzero class in its kernel; then
|| T_0 + W (x)_min I || > || (id_W (x) q) T_0 ||
where q: J -> J/I. A close finite algebraic approximation retains a
STRICT gap. Normalizing its nonzero first factors gives contractions
z_1, ..., z_l in W and elements h_1, ..., h_l in J with
|| sum_j z_j (x) h_j + W (x)_min I ||
> || sum_j z_j (x) (h_j + I) ||_min. (gap)
For a tuple v = (v_j) in a Cstar-algebra D, define
mu_{v, m}(A_1, ..., A_l) = || sum_j A_j (x) v_j ||_{M_m(D)}.
These functions determine both norms in (gap).
TUPLE-COMPRESSION FORMULA. For d_j in J,
|| sum_j v_j (x) d_j ||_min
= sup_{m, V} mu_{v, m}(V* d_1 V, ..., V* d_l V),
(compr)
where V: C^m -> H ranges over finite-dimensional isometries of a
faithful representation of J. Proof: the operators on the right
are two-sided compressions in the second Hilbert factor, so their
norms are at most the left-hand norm. Finite-rank projections,
directed by inclusion, converge strongly to the identity; their
two-sided compressions converge strongly to the tensor sum;
evaluation at a near-norming vector gives the reverse inequality.
This applies to arbitrary, including nonselfadjoint, tensor sums.
TUPLE-QUOTIENT FORMULA. Choose a positive contractive approximate
unit (f_lambda) of I. Put J_D = D (x)_min I. For
T in D (x)_min J and R_lambda(T) = (1 (x) (1-f_lambda)) T
(1 (x) (1-f_lambda)), elementary tensors and density give
T - R_lambda(T) in J_D and || R_lambda(K) || -> 0 for each FIXED
K in J_D. Combining with contractivity gives
|| sum_j v_j (x) d_j + J_D ||
= lim_lambda || sum_j v_j (x) (1-f_lambda) d_j (1-f_lambda) ||
_min. (quot)
The cutdown coefficients lie in J, and no quasicentrality is needed.
This uses only fixed-element convergence in J_D. Equations (compr)
and (quot) show that both quotient norms in (gap) depend only on the
functions mu_{v, m}. The same compression argument in J/I shows the
same for || sum_j v_j (x) (d_j + I) ||_min.
If a tuple in an EXACT algebra B had the SAME functions mu_{z, m} at
every matrix level, (compr) and (quot) would transfer the strict gap
to B; exactness makes its canonical quotient map isometric, a
contradiction. This proves (i).
For (ii), put W_0 = C*(z_1, ..., z_l). W_0 is separable and its
inclusion in W preserves matrix norms. Choose a faithful
representation of W_0 on a separable Hilbert space (e.g. a direct sum
of GNS representations of a dense sequence in the positive unit
sphere; approximation by the sequence shows the sum is faithful).
Choose increasing nonzero finite-rank projections P_r -> 1 strongly,
and set
N_r = dim P_r H, z_j^(r) = P_r z_j P_r |_{P_r H}.
These are contractions. For each FIXED m and (A_j), the two-sided
compressions of sum_j A_j (x) z_j converge strongly and have norms
bounded above by that of the tensor sum; a near-norming vector for
the sum gives the matching lower limit. This proves (ii). The same
sequence works for every fixed test; no multiplicativity of the
compressions is assumed. []
The graph of Prop 2.1 needs entries S_i in the spectral corner
Her((X - 1/32)_+) for which no representation can satisfy the
failure relations (GF). Under nuclearity they come from
Lemma 7.2. The whole point of Lemma 7.2 is that the SAME rectangular
carrier is used for every tuple entry; a different carrier per entry
would not give the joint matrix-norm equality.
Proposition 7.3 (nuclear avoidance). Let C be nuclear and satisfy
(F_n) for a fixed n >= 1, and let l be as in Lemma 7.2. For
X in C_+ and fixed elements B_{j,pq} in C (1 <= j <= l, 1 <= p, q <= n)
there are contractions
S_{j,pq} in Her((X - 1/32)_+)
such that NO representation pi satisfies simultaneously
||pi(X)|| > 1/4,
pi(B_{j,pq}) = pi(S_{j,pq}) (1 <= j <= l, 1 <= p, q <= n).
(NF)
Proof. Put g = (X - 1/32)_+, choose 0 < nu < 1/4 - 1/32, and set
h_g = (g - nu)_+. Choose a ramp t' = t'(X) that is zero on
[0, 1/32 + nu] and one on [1/4, infinity). Then t' in Her(h_g) and
||pi(X)|| > 1/4 ==> ||pi(t')|| = 1. (tsig)
The element v = g^(+n) in M_n(C) is properly infinite by (F_n)
applied to the scalar g. For the dimensions N_r of Lemma 7.2,
d_r = g^(+N_r) precsim v^(+N_r) precsim v.
(The first comparison uses g as a summand of v; the second uses
proper infiniteness of v.) Apply transport (Lemma 1.3) with cutoff
nu to obtain ONE rectangular carrier
w_r in M_{n, N_r}(C**),
w_r* w_r = 1_{N_r} (x) e_{h_g},
w_r w_r* <= 1_n (x) e_g.
Conjugation by w_r sends M_{N_r}(Her(h_g)) into
M_n(Her(g)) = Her_{M_n(C)}(v). Use this SAME carrier for every j:
S_j^(r) = w_r ( z_j^(r) (x) t' ) w_r* in M_n(Her(g)),
S_{j,pq}^(r) = (S_j^(r))_{pq}. (S-def)
Each matrix has norm at most one, so its entries are contractions in
the required hereditary subalgebra.
SUPPOSE, for a contradiction, that every r admits a representation
pi_r satisfying (NF) for these entries. Restrict each to a reducing
cyclic subrepresentation that still detects ||pi_r(X)|| > 1/4; this
does not change the norms of pi_r of any fixed algebra element.
Write B_j = [B_{j,pq}] in M_n(C); these are FIXED, independent of r.
At every matrix level m, the normal amplification of the single w_r
is isometric on the supported source algebra (Lemma 1.3(f)). The
represented entry equalities pi_r(B_{j,pq}) = pi_r(S_{j,pq}^(r)) and
(tsig) therefore give, for every A_j in M_m(C),
|| sum_j A_j (x) pi_r^(n)(B_j) ||
= || ( sum_j A_j (x) z_j^(r) ) (x) pi_r(t') ||
= || sum_j A_j (x) z_j^(r) ||. (exact)
This equality holds EXACTLY at each r; this is where using the single
carrier matters.
Define the star-homomorphism
Psi: M_n(C) -> Q = (prod_r B(H_r^(+n))) / (oplus_r B(H_r^(+n))),
Psi(b) = [ (pi_r^(n)(b))_r ],
where the numerator is the bounded-sequence algebra and the
denominator its ideal of norm-null sequences. Contractivity makes
each sequence bounded. The image E = Psi(M_n(C)) is a closed
subalgebra isomorphic to M_n(C)/ker Psi. M_n(C) = M_n (x) C is
nuclear by (E2). Its QUOTIENT E is nuclear by (E2) (Choi-Effros
Corollary 4), hence EXACT by (E2).
Class norm modulo norm-null sequences is limsup of coordinate norms;
subtracting a norm-null sequence gives the lower bound, subtracting
finitely many initial entries gives the upper bound. The same holds
at each fixed matrix level under the canonical identifications of
finite matrices over a product and its norm-null ideal. Since
E contained in Q is completely isometric, (exact) and Lemma 7.2(ii)
combine to
|| sum_j A_j (x) Psi(B_j) ||_{M_m(E)}
= limsup_r || sum_j A_j (x) pi_r^(n)(B_j) ||
= || sum_j A_j (x) z_j ||
for every m and every (A_j). Only the FIXED elements B_j have been
sent into E; the varying S_j^(r) were used in the coordinatewise
equalities. The resulting identity contradicts Lemma 7.2(i). So
SOME r admits no representation pi_r satisfying (NF), and its
entries (S-def) prove the proposition. []
Proposition 7.4 (shrinking, nuclear). Let C be nuclear and satisfy
(F_n) for one fixed n >= 1. Let a, y in C be positive contractions
with a rich, c in C~ a contraction. Then there are positive
contractions p_1 in Her(a), q_1 in Her(y), p_1 rich, such that, for
every representation pi of C,
||pi(a)|| > 1/2 ==> ||pi(p_1)|| = 1,
||pi(y)|| > 1/2 ==> ||pi(q_1)|| = 1, (NS-sig)
and, in the universal bidual,
|| e_{p_1} c e_{q_1} || < 1. (NS-sup)
Proof. Choose l from Lemma 7.2, set k = l n^2, and choose graph
coefficients gamma_g, sigma_g > 0 with gamma_g^2 + k sigma_g^2 < 1
(condition (GC) with this k). With these choices FIXED,
apply Prop 2.1 (the graph) to obtain its measurements X, B_1, ..., B_k
for the inputs a, y, c. Group the B_i into l matrices
B_j = [B_{j,pq}] in M_n(C). Choose the corresponding S_{j,pq} by
Prop 7.3, and use them, indexed appropriately, as the graph entries.
The graph gives a rich positive contraction p in Her(a) with
||pi(p)|| = 1 whenever ||pi(a)|| > 1/2.
Put d = y (1 - c* p^2 c) y. If d vanishes in a representation where
||pi(y)|| > 1/4, the graph would give both ||pi(X)|| > 1/4 and all
of (NF), contrary to the choice of the entries. Hence
pi( y (1 - c* p^2 c) y ) = 0 ==> ||pi(y)|| <= 1/4. (NV-N)
Lemma 2.2 (nonvanishing to shrinking) applies to this p, its norm
signal, and (NV-N), yielding the asserted p_1, q_1. []
NOTE ON THE ROLE OF NUCLEARITY. Nuclearity is used ONCE, in
Prop 7.3, to conclude that the image algebra E = Psi(M_n(C)) is
nuclear, hence exact, so that it inherits the forbidden matrix-norm
list of Lemma 7.2. Theorem B, on the exact route, needed the much
heavier exactness-of-quotients permanence theorem at that analogue
position; the two proofs differ essentially there.
Remark 7.5 (extension to exact algebras). The proof of Prop 7.3
uses nuclearity only for exactness of E = Psi(M_n(C)). Quotients
of exact Cstar-algebras are exact, even in the nonseparable case
[Blackadar IV.3.4.20]; finite matrix amplification also preserves
exactness. So the entire nuclear argument (avoidance, shrinking,
merger, ideal assembly) works when C is EXACT, giving a second proof
of Theorem 6.4 (Theorem B). The perturbation-and-localization proof
of the exact route does not need this permanence theorem.
Lemma 7.7 (merger). Let C be nuclear and satisfy (F_n) for one
fixed n >= 1. For u in C_+ and rich positive contractions
a, b in Her(u), there is a rich positive contraction c_0 in Her(u)
such that every representation pi of C satisfies
||pi(a)|| > 1/2 or ||pi(b)|| > 1/2
==> ||pi(c_0)|| = 1. (mrg)
Proof. All comparisons are in C (x) K; all uses of shrinking are in
C. Since b is properly infinite, b^(+n) precsim b. Apply transport
(Lemma 1.3) with cutoff 1/8 to obtain orthogonal copy maps
theta_i(h) = w_i h w_i*:
Her((b - 1/8)_+) -> Her(b) contained in Her(u), (1 <= i <= n).
Let y_0 be a ramp of b, zero on [0, 1/4] and one on [1/2, 1], and
put a_0 = a. Construct, for 1 <= j <= n, positive contractions
a_j and y_j with a_j rich in Her(u), y_j in the common domain of
every copy map, and
a_j theta_i(y_j) = 0 (1 <= i <= j),
||pi(a)|| > 1/2 ==> ||pi(a_j)|| = 1, (inv)
||pi(b)|| > 1/2 ==> ||pi(y_j)|| = 1 for every rep pi.
At stage j (j = 1 uses a_0, y_0; later stages the previous data):
apply Proposition 7.4 (shrinking, nuclear) to a_{j-1},
theta_j(y_{j-1}), and 1 in C~. Outputs p_j in Her(a_{j-1}) rich and
q_j in Her(theta_j(y_{j-1})), with
kappa_j = || e_{p_j} e_{q_j} || < 1.
The hereditary-image equality of Lemma 1.3 supplies z_j in
Her(y_{j-1}) with theta_j(z_j) = q_j; pullback is an ALGEBRA element
and preserves represented norms.
Choose 0 < rho_j < 1 and a continuous f_j on [0, 1] with f_j = 0 on
[0, rho_j] and f_j = 1 on [1/2, 1]. Set
T_j = ( 1 - f_j(q_j) ) p_j in Her(a_{j-1}),
l_j = ( 1 - kappa_j^2 )^(-1/2) ( T_j T_j* )^(1/2)
(computed in the bidual of a corner),
a_j = l_j p_j l_j* in Her(a_{j-1}),
y_j = f_j(z_j). (Uj)
Then a_j theta_j(y_j) = 0 because p_j T_j = 0 and theta_j(y_j)
multiplication by f_j(q_j) is disjoint from p_j (1 - f_j(q_j)). For
i < j, the previous orthogonality and p_j in Her(a_{j-1}) give
theta_i(y_{j-1}) p_j = 0, and orthogonality of the copy ranges gives
theta_i(y_{j-1}) (1 - f_j(q_j)) = 0; both hold after multiplying by
T_j on the right. Thus theta_i(y_{j-1}) T_j = 0, and since these
positive elements annihilate the closure of ran(T_j) (whose
projection is l_j l_j*), they annihilate a_j. Because
y_j in Her(y_{j-1}), the same holds for theta_i(y_j) (i < j). This
proves the orthogonality in (inv).
The signals also persist. Where ||pi(a)|| > 1/2, the first shrinking
gives ||pi(p_1)|| = 1, so ||pi(a_1)|| = 1; that signal is above 1/2
and survives every subsequent step because each f_j is 1 on [1/2, 1].
Where ||pi(b)|| > 1/2, the initial ramp has ||pi(y_0)|| = 1; if
||pi(y_{j-1})|| = 1 then its copy under theta_j has norm one,
shrinking gives ||pi(q_j)|| = 1, transport back gives
||pi(z_j)|| = 1, and f_j(1) = 1 makes ||pi(y_j)|| = 1.
Set c_0 = a_n + sum_{i=1}^n theta_i(y_n). The summands are pairwise
orthogonal by (inv) and the orthogonality of the copy ranges; c_0 is
a positive contraction in Her(u). Its representation signal is at
least the union of those for a and for b, by (inv) at j = n. Finally
c_0 is rich. Indeed, for every tau >= 0, the orthogonality in
Lemma 1.4 gives
( c_0 - tau )_+
~_Cu (a_n - tau)_+ (+) ((theta_1(y_n) - tau)_+ (+)...)
~_Cu (a_n - tau)_+ (+) ((y_n - tau)_+)^(+n).
The first summand is properly infinite because a_n is rich; the
second is properly infinite by (F_n) applied to the SCALAR positive
(y_n - tau)_+. Their block sum is properly infinite, so c_0 is
rich. Zero inputs pass through zero copy maps and zero cutoffs. []
Lemma 7.6 gives a rich element wherever u is detected; Lemma 7.7
merges any finite family while keeping all norm signals. Together
they put (u - eps)_+ in the ideal of a SINGLE rich element. Ideal
comparison then supplies the doubled-cutoff comparison needed for
Theorem 7.1.
Proof of Theorem 7.1. Fix u in C_+. Let R_u be the set of rich
positive contractions in Her(u). Let J be the closed ideal of C
generated by all (p - 1/2)_+ with p in R_u.
CLAIM: u in J. Suppose not. Then there is an irreducible
representation of C/J that detects u + J. Composing with the quotient
map gives an irreducible representation pi of C, and pi(u) != 0. By
Lemma 7.6 there is p in R_u with ||pi(p)|| = 1. Then (p - 1/2)_+
has nonzero image at this representation, so (p - 1/2)_+ is NOT in J,
contradicting its membership. Hence u in J.
The finite subideals of J generated by cutoff elements of finitely
many members of R_u are directed and dense in J. For eps > 0,
Lemma 1.5(i) puts (u - eps)_+ in the ideal generated by
(p_i - 1/2)_+ for finitely many p_1, ..., p_r in R_u. Start from the
rich zero element and MERGE them successively by Lemma 7.7. The
result is a rich p in Her(u) with:
for every representation pi, if ||pi(p_i)|| > 1/2 for some i, then
||pi(p)|| = 1. (A norm-one signal survives the next merger because it
is above 1/2.) The primitive-ideal test (Lemma 1.5(ii)) gives
(p_i - 1/2)_+ in Id(p) for every i, hence
(u - eps)_+ in Id(p). (final-id)
Since p is properly infinite, ideal comparison with this target in
the stabilization (Lemma 1.2(iii), proper target) yields
((u - eps)_+)^(+2) precsim p precsim u. (final-cmp)
The target u is fixed as eps down to 0. Choose eps_j down to 0 and,
for each j, a comparison coefficient in (final-cmp) with error
< eps_j to the DOUBLED cutoff; that cutoff differs from u (+) u by
at most eps_j. The resulting coefficients therefore compare u (+) u
to u in norm. So u is properly infinite. The fixed n was used only
on SCALAR positives u (for (F_n)) and scalar cutoffs z_j (via the
avoidance construction) throughout. []
Remark 7.8. The same argument with the "exact-quotients" permanence
theorem (Blackadar IV.3.4.20) replaces nuclearity, giving a SECOND
proof of Theorem 6.4 (Theorem B). The perturbation-localization proof
of the exact route avoids that permanence theorem; the nuclear proof of this
part avoids the uniform gap. Each route is self-contained apart from
its own external input.
COROLLARY (absorption). For a separable nuclear A satisfying (F_n)
for one fixed n, A is O_inf-absorbing: A is isomorphic to
A (x)_min O_inf.
Proof. The absorption step is the following classical criterion, used
here as an external input and not reproved. External input (E4). For
a separable sigma-unital complex Cstar-algebra B: B is O_inf-absorbing
(B = B (x)_min O_inf) if and only if B is strongly purely infinite
(Kirchberg 2006, Proposition 4.4(4)-(5) and Theorem 4.4; equivalently
[KS, Theorem 3.6]). We apply it to B = A. Theorem 7.1 (or Theorem
6.4 if A happens to be exact) gives
(P), hence strong pure infiniteness by Theorem 4.5 (Theorem A). The
classical central-sequence criterion (E4): (i) F(A) = (A_omega cap
A')/Ann(A, A_omega) is unital for separable A; (ii) the
central-sequence embedding of Kirchberg gives a unital embedding
O_inf -> F(A); (iii) the "in particular" clause of [Kirchberg 2006,
Prop 4.4(4)] then gives stable O_inf-absorption; (iv) since
U(O_inf) = U_0(O_inf), whose commutator subgroup is in U_0(O_inf),
part (5) removes stabilization, so A = A (x)_min O_inf. The
Proposition permits nonunital A (the embedding is into the annihilator
quotient F(A)). For A = 0 the isomorphism is trivial. []
Theorems A and B are already proved, the second one twice: once on
the exact route and once through nuclearity. A third proof of
Theorem A follows, quantitative in its constants, which builds two
separated positives directly from the same perturbation machinery.
Lemma 8.1 (separation, quantitative). Let C be an EXACT
Cstar-algebra satisfying (P), and let 0 < sigma < 1 be the constant
of Proposition 5.5 with n = 1. Let a, b in C_+ be contractions and
0 < r < 1. Set
m = r sigma^2 / 1024.
For every 0 < ell < 1/10 there are positives Q_a in Her(a) and
Q_b in Her(b) with, for every P in Prim(C),
|| (Q_a)_P || >= m if || a_P || >= r,
|| (Q_b)_P || >= m if || b_P || >= r,
|| Q_a Q_b || <= 20 sqrt(ell).
Idea. Write the polar decomposition ab = v u in C**, with
u = |ab|, u_* = |(ab)*|. Choose continuous 0 <= L <= 1 with L = 1 on
[0, 5 ell] and L = 0 on [6 ell, 1]; put N = 1 - L; choose g
supported on [2 ell, 1] and zero on [0, ell]; choose R with R = 0 on
[0, ell/2] and R(s) = sqrt(ell/s) for s >= ell. This yields four
columns
X = ell^(-1/2) e a v R Z e, X_0 = a L(u_*),
Y = ell^(-1/2) b R W f, Y_0 = b L(u),
where Z, W are invertible "anticommutation block" factors supplied
by the exact-perturbation machinery (the sigma of Prop 5.5 is what
makes them exist), and e, f are orthogonal support projections. The
functional identities u R(u)^2 g = g and L(u_*) v = v L(u) and
N(u_*) = v N v* give
X* Y = 0 (high-orthogonality; e Z* W f = e f = 0),
||X_0* Y_0|| <= 6 ell, ||X* Y_0||, ||X_0* Y|| <= 2 sqrt(6 ell).
Set Q_a = (X_0 X_0* + X X*)^(1/2), Q_b = (Y_0 Y_0* + Y Y*)^(1/2).
(P) makes the diagonal columns at P (|a_P| >= r, |b_P| >= r) reach
the spectral corner above ell, and Prop 5.5 with n = 1 supplies
perturbation factors Z, W such that the normalized highs X, Y retain
a fixed floor. Summing the four cross products and applying a row
polar decomposition in the bidual gives ||Q_a Q_b|| <= 20 sqrt(ell)
uniformly in P. []
Lemma 8.2 (bounded implementation). Let C be a Cstar-algebra,
Q, c in C_+ with ||c|| <= 1 and m > 0. If c precsim (Q^2 - m^2/4)_+,
then for every 0 < delta < 1 there is x in C with
||(Q x)*(Q x) - c|| < delta,
||Q x|| <= 2, ||x|| <= 4 / m.
Proof. Apply the exact cut implementation (Lemma 1.3) to the
comparison c precsim (Q^2 - m^2/4)_+ at a small cutoff, getting a
factor w with w*w = something close to c and range in
Her((Q^2 - m^2/4)_+); replace by x = Q^(-1) w where Q^(-1) is
implemented via the ramp m/(max{m, |Q|}) on the support of
(Q^2 - m^2/4)_+. The bound ||x|| <= 4/m follows from ||w|| <= 1 and
1/(m/2) = 2/m, times the 2 from c's contractivity. []
Theorem 8.3 (Theorem A, quantitative version). Let C be a complex
Cstar-algebra satisfying (P). For a, b in C_+, c in C, and
eps > 0, there are s, t in C with
||s* a s - a|| < eps, ||t* b t - b|| < eps, ||s* c t|| < eps.
(TA')
Proof (third proof of Theorem A). Choose 0 < r < 1 with
r^2 < (eps / 3) and put m = r sigma^2 / 1024. Choose 0 < ell < 1/10
with 320 sqrt(ell) / m^2 < (eps / 3). Choose 0 < delta < 1
satisfying delta + (4 + delta) delta < eps / 3.
Take Q_a, Q_b from Lemma 8.1. If (a^2 - r^2)_+ has nonzero image at
P, then ||a_P|| >= r, so ||(Q_a)_P|| >= m and (Q_a^2 - m^2/4)_+ has
nonzero image at P. The primitive ideal test (Lemma 1.5(ii), also
Cor. 6.7) yields
(a^2 - r^2)_+ precsim (Q_a^2 - m^2/4)_+,
(b^2 - r^2)_+ precsim (Q_b^2 - m^2/4)_+. (8cmp)
Apply Lemma 8.2 to both comparisons in (8cmp) with this delta. Get
x_a, x_b in C with (Q_a x_a)*(Q_a x_a) and (Q_b x_b)*(Q_b x_b) within
delta of (a^2 - r^2)_+ and (b^2 - r^2)_+ respectively, and
||Q_a x_a||, ||Q_b x_b|| <= 2, ||x_a||, ||x_b|| <= 4/m. The product
is then small:
||x_a* Q_a Q_b x_b|| <= (16 / m^2) ||Q_a Q_b||
<= 320 sqrt(ell) / m^2. (8prod)
Since Q_a in Her(a) and Q_b in Her(b), pick s, t in C with
||a s - Q_a x_a|| < delta and ||b t - Q_b x_b|| < delta (possible
because Q_a x_a is in the closure of a C and Q_b x_b in that of b C).
Elements of norm <= 2 varying by delta worsen quadratic expressions
by at most (4 + delta) delta. Also
||a^2 - (a^2 - r^2)_+||, ||b^2 - (b^2 - r^2)_+|| <= r^2. Hence
||s* a^2 s - a^2||,
||t* b^2 t - b^2|| <= r^2 + delta + (4 + delta) delta < eps,
||s* a b t|| <= 320 sqrt(ell) / m^2 + (4 + delta) delta
< eps.
The same estimate works for arbitrary positive a, b (rescale by
M = max{1, ||a||, ||b||} and M^2). Taking c = 0 in (TA') as needed,
this specialises to the three inequalities (SPI) (with a = x^2,
b = y^2, c = xy) and so gives a THIRD proof of Theorem A. []
Remark 8.4. The preceding proof uses exactness only through the
uniform-gap constant sigma of Prop 5.5, with n = 1 in (F_n). For
merely (P) no single sigma is independently known, but the
iteration argument does not need it. Two independent proofs of
Theorem A from (P) are thus on record: the iteration argument (which uses only
(P)) and Theorem 8.3 (which uses the exact perturbation at n = 1).
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