KADISON'S SIMILARITY THEOREM
Through Uniform Derivation Estimates
A COMPLETE PROOF, WRITTEN FOR A CURIOUS BEGINNER
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WHAT THIS DOCUMENT IS
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Kadison's similarity problem (1955) asks whether every bounded
unital homomorphism from a unital C*-algebra into the bounded
operators on a Hilbert space is similar, by a bounded change of
coordinates, to a *-homomorphism. This document proves that the
answer is yes. Along the way it proves that every von Neumann
algebra is hyperreflexive with one universal constant.
Nothing here is a survey. Every lemma needed for the main
theorems is stated and proved. Where the proof uses a
well-established result from the literature, that result is
named precisely (Paulsen, Kirchberg, Christensen, Ringrose,
Dickson, Popa, Haagerup, Schur-Weyl, Kaplansky density, Arveson)
so the reader can look it up. Background material is squeezed
into the few paragraphs that the main argument actually uses.
Read the document twice. First pass: read the boxed statements;
you will see the plot. Second pass: read the proofs and check
each estimate. A high-school reader who knows what a vector
space is can follow the first pass fully; the second pass needs
a teacher nearby but nothing beyond freshman linear algebra and
the willingness to accept the named black boxes.
Conventions. Base field C. Hilbert spaces are complex. Inner
products <x,y> are linear in x (first slot), conjugate-linear
in y. Every algebra homomorphism is complex-linear and unital.
The adjoint of T is T^*, so <T^* x, y> = <x, T y>. For operators
T, S on a Hilbert space, [T,S] = TS - ST.
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PART 0. THE THREE THEOREMS
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Every statement below is proved in this document.
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THEOREM A (Similarity; Kadison 1955, resolved here).
Let A be a unital C*-algebra, H a complex Hilbert space, and
pi : A -> B(H) a bounded unital algebra homomorphism. There is
an invertible bounded operator S on H with bounded inverse such
that a |-> S pi(a) S^{-1} is a *-homomorphism. In particular
S pi(a^*) S^{-1} = (S pi(a) S^{-1})^* for every a in A.
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THEOREM B (Universal hyperreflexivity).
Let M be a unital von Neumann algebra in B(H). Define the
off-diagonal distance seminorm of T in B(H) by
alpha_M(T) = sup { ||(1 - e) T e|| : e a projection in M' }.
Then the distance dist(T, M) = inf { ||T - a|| : a in M } obeys
dist(T, M) <= 2 C alpha_M(T) for all T in B(H),
with the SAME absolute constant C from Theorem C below, for
every M and every H.
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Both theorems come from one inequality:
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THEOREM C (Uniform commutator estimate; the central estimate).
There is a finite constant C such that, for every complex Hilbert
space K, every unital von Neumann algebra P in B(K), every Y in
B(K), every integer h >= 1, and every X in M_h(P),
||[Y^(h), X]|| <= C g_P(Y) ||X||,
where g_P(Y) = sup { ||[Y,a]|| : a in P, ||a|| <= 1 } and Y^(h)
denotes the diagonal amplification of Y to K^h.
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Here and below, M_h(P) is the algebra of h x h matrices with
entries in P, acting entrywise on K^h.
How to read Theorem C. g_P(Y) is a scalar: how large a commutator
Y makes with elements of P at matrix size one. The claim is that
the SAME scalar bounds the commutator of Y with whole h x h
matrices of elements of P, uniformly in h = 1, 2, 3, ...
For h = 1 the inequality is the definition of g_P and holds
with C = 1. The content is the uniformity in h. If C grew with h,
the whole scheme would collapse. It does not.
Dispatch. Parts 1-3 set up the language, prove the toolbox, and
state the reduction plan. Parts 4-10 prove the free-product
corner estimate, the technical heart. Part 11 places it in
ultrapowers and yields Theorem C in II_1 factors. Part 12
extends Theorem C to arbitrary von Neumann algebras and derives
Theorems A and B.
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PART 1. WHAT IS IN PLAY
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A HILBERT SPACE K is a complete complex vector space with an
inner product. B(K) is the algebra of bounded linear operators
K -> K, with ||T|| = sup { ||T x|| : ||x|| <= 1 } < infinity.
The adjoint T^* of T is defined by <T^* x, y> = <x, T y>. The
involution T |-> T^* is conjugate-linear and reverses products:
(TS)^* = S^* T^*.
A C*-ALGEBRA A is a *-closed, norm-closed unital subalgebra of
some B(K). Equivalently (Gelfand-Naimark), an abstract algebra
satisfying the axioms ||a^* a|| = ||a||^2 etc. We work concretely
and lose nothing.
A VON NEUMANN ALGEBRA M is a unital *-subalgebra of B(K) closed
in the strong operator topology (T_n -> T iff T_n x -> T x in
norm for every x in K). Equivalently, M = M'' := (M')'' where
M' = { S : S T = T S for every T in M }. Von Neumann algebras
have plenty of projections: p = p^* = p^2.
A FACTOR is a von Neumann algebra with trivial center,
Z(M) = M intersect M' = C 1.
A II_1 FACTOR is an infinite-dimensional factor with a
FAITHFUL NORMAL TRACE tau : M -> C. Faithful: tau(T^* T)=0 iff
T=0. Normal: strong-operator continuous on the unit ball. Tracial:
tau(TS) = tau(ST). Unital: tau(1) = 1. Infinite-dimensional plus
tracial forces type II_1.
Every finite von Neumann algebra carries an L^2 COMPLETION:
L^2(M) is the completion of M under the norm ||x||_2 = tau(x^*x)^{1/2}.
Left multiplication L_x : y |-> xy extends to a bounded operator
on L^2(M); the map x |-> L_x is a faithful *-representation
called the STANDARD REPRESENTATION.
A CORNER of M at a projection e is eMe = { exe : x in M }. It is
a *-subalgebra with unit e, is itself a von Neumann algebra, and
carries the normalized trace tau_e(x) = tau(x)/tau(e).
A PARTIAL ISOMETRY is v with v^* v and vv^* projections (the
SOURCE and TARGET). An ISOMETRY has v^* v = 1. In a II_1 factor
M, projections of equal trace are unitarily conjugate, and every
projection of smaller trace fits under a projection of larger
trace.
A SYMMETRY is s = s^* = s^{-1}. A HAAR UNITARY in a diffuse
abelian subalgebra A of M is a unitary u generating A; the powers
u^n (n in Z) are orthonormal in L^2(M), hence converge weakly
to 0 as n -> +-infinity.
Conventions for inner products on L^2(M): <x,y> = tau(y^* x). This
makes inner products linear in the first variable. For right
multiplication, R_a xi = xi a; note R_a R_b = R_{ba} (NOT R_{ab}).
The tracial commutation theorem: L(M)' = { R_a : a in M }.
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PART 2. DERIVATIONS AND COMPLETE BOUNDEDNESS
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2.1 Derivations.
Let sigma : A -> B(K) be a unital *-representation. A
SIGMA-DERIVATION is a linear map Delta : A -> B(K) satisfying
the product rule
Delta(ab) = Delta(a) sigma(b) + sigma(a) Delta(b).
Delta is INNER if there is V in B(K) with
Delta(a) = V sigma(a) - sigma(a) V.
Unitality forces Delta(1) = 0: for a = b = 1 the rule gives
Delta(1) = Delta(1) + Delta(1).
RINGROSE (1972). Every derivation from a C*-algebra into a
Banach bimodule is bounded. In particular, "bounded derivation"
is a redundant hypothesis; every derivation into B(K) is
automatically bounded. We use this only to phrase the argument
in terms of bounded derivations without loss.
2.2 Complete boundedness, row and column norms.
For u : A -> B(F) linear and h >= 1, the h-th amplification
u_h : M_h(A) -> M_h(B(F)) acts entrywise:
u_h( (a_{ij})_{i,j=1..h} ) = ( u(a_{ij}) )_{i,j=1..h}.
Define
||u||_h = operator norm of u_h,
||u||_cb = sup_{h >= 1} ||u||_h in [0, +infinity].
u is COMPLETELY BOUNDED if ||u||_cb < infinity. Every unital
*-homomorphism is completely contractive: ||u||_cb <= 1.
For a bounded unital phi : A -> B(K), define also
||phi||_row = sup_h ||phi_{1,h}||,
||phi||_col = sup_h ||phi_{h,1}||.
DICKSON'S ROW LEMMA (2014, Lemma 2.9). Every bounded unital
homomorphism phi : A -> B(K) obeys
||phi||_row <= sqrt(2) ||phi||^2.
Applying the same estimate to phi^#(a) = phi(a^*)^* (which is
again a unital bounded homomorphism, of the same norm) gives the
same bound for the column norm:
||phi||_col <= sqrt(2) ||phi||^2.
Consequently, if ||phi|| <= 2, then ||phi||_row, ||phi||_col <=
4 sqrt(2) =: c_row.
The proof of Dickson's lemma sits on Haagerup's little
Grothendieck inequality; we take it as a black box.
2.3 Two named black boxes.
PAULSEN'S THEOREM (1984, p.~226). For a unital bounded
homomorphism pi : A -> B(H), the following are equivalent:
(i) pi is similar to a *-homomorphism;
(ii) ||pi||_cb < infinity.
In that case the OPTIMAL similarity condition number,
inf { ||S|| ||S^{-1}|| : S pi S^{-1} is a *-homomorphism },
equals ||pi||_cb, and the resulting *-homomorphism may be
taken completely contractive.
CHRISTENSEN'S DERIVATION THEOREM (1977). A derivation Delta :
A -> B(K) into a faithful *-representation is inner iff it is
completely bounded.
2.4 The rectangular toolkit.
A RECTANGULAR DERIVATION for two unital *-representations
sigma : A -> B(K) and lambda : A -> B(F) is a linear map
Delta : A -> B(K,F) with
Delta(ab) = Delta(a) sigma(b) + lambda(a) Delta(b).
It obeys Delta(1) = 0.
LEMMA 2.1 (Row and column bounds).
Every bounded rectangular derivation Delta satisfies
||Delta||_row <= c_row ||Delta||,
||Delta||_col <= c_row ||Delta||,
where ||.||_row and ||.||_col are the rectangular amplifications
sup_p ||Delta_{1,p}||, sup_q ||Delta_{q,1}||, and c_row = 4 sqrt(2)
is as in 2.2.
Proof. Trivial if Delta = 0. Let d = ||Delta|| > 0. Consider the
upper-triangular map
Phi(a) = [[ lambda(a), d^{-1} Delta(a) ],
[ 0, sigma(a) ]] in B(F + K).
The derivation identity says precisely that Phi is multiplicative.
Diagonal is contractive, the corner has norm 1, so ||Phi|| <= 2.
The off-diagonal compression of Phi_{p,q}, from K^q to F^p, is
d^{-1} Delta_{p,q}. Dickson's row estimate applied to Phi (||Phi||
<= 2) gives ||Phi_{1,p}|| <= c_row, i.e. ||Delta_{1,p}|| <= d c_row.
Same for columns. QED
LEMMA 2.2 (Completely bounded rectangular derivations implement).
Let sigma, lambda, Delta be as in 2.4. If Delta is completely
bounded with d = ||Delta|| > 0, there is V in B(K,F) with
Delta(a) = V sigma(a) - lambda(a) V for all a in A,
||V|| <= d (1 + ||Delta||_cb/d)^2.
Proof. Use Phi : A -> B(F + K) from Lemma 2.1. Phi is cb with
||Phi||_cb <= 1 + ||Delta||_cb/d =: G.
Paulsen supplies S invertible with D = S^* S obeying
||D|| ||D^{-1}|| <= ||S||^2 ||S^{-1}||^2 <= G^2,
and D Phi(a) = Phi(a^*)^* D for all a. Write D = [[D_11, D_12],
[D_21, D_22]] on F + K. The (1,1) entry of the intertwining
identity reads D_11 lambda(a) = lambda(a) D_11. The (1,2) entry:
d^{-1} D_11 Delta(a) + D_12 sigma(a) = lambda(a) D_12.
Since D >= ||D^{-1}||^{-1} I, the compression D_11 is positive
invertible, D_11^{-1} commutes with lambda(A), and
||D_11^{-1}|| <= ||D^{-1}||. Set V = -d D_11^{-1} D_12. Then
Delta(a) = d D_11^{-1} lambda(a) D_12 - d D_11^{-1} D_12 sigma(a)
= -lambda(a) V + V sigma(a).
And ||V|| <= d ||D_11^{-1}|| ||D_12|| <= d ||D^{-1}|| ||D|| <= d G^2.
QED
2.5 Implementation on a cyclic domain.
Let sigma : A -> B(K) be a *-representation. A vector zeta in K
is CYCLIC for sigma if the set { sigma(a) zeta : a in A } is
norm-dense in K.
LEMMA 2.3 (Column factorization in a cyclic representation).
Let P in B(K) be a unital von Neumann algebra with a cyclic
vector Omega. Then for every h >= 1 and every unit vector
xi = (xi_1, ..., xi_h) in K^h, there are a unit vector zeta in K
and a contraction v = (v_1, ..., v_h) in M_{h,1}(P) with
v zeta = xi.
Proof. Put N = P'. Since P has a cyclic vector Omega, the same
Omega SEPARATES N: if x in N and x Omega = 0, then for every a in P,
x a Omega = a x Omega = 0,
so x vanishes on the dense set P Omega, so x = 0. Let
L = closure(N Omega) in K. This subspace reduces N, and the
restriction r : N -> B(L) is a faithful normal *-homomorphism
onto a von Neumann algebra. Its inverse is also normal. Omega is
cyclic and separating for r(N).
By Haagerup's standard form theorem (1975, Lemma 2.10(1)) the
normal positive functional
omega(x) = sum_{i=1}^h <x xi_i, xi_i>, x in N,
is a vector functional for the representation r: there is zeta
in the natural cone of r(N) with <r(x) zeta, zeta> = omega(x) for
every x in N; in particular ||zeta||^2 = omega(1) = 1. Through
the normal isomorphism r, <x zeta, zeta> = omega(x) for all x in N.
The map x zeta |-> (x xi_1, ..., x xi_h) is isometric, since both
squared norms equal omega(x^* x). Extend it by zero on
L-overline{N zeta} to a contraction v : K -> K^h. Because the
subspace N zeta reduces N and the map intertwines N-actions,
v x = x^{(h)} v for every x in N. Each coordinate v_i lies in
N' = P, and v zeta = xi. QED
LEMMA 2.4 (Implementation on a cyclic domain).
Let sigma : A -> B(K) and lambda : A -> B(F) be unital *-reps,
with sigma cyclic. Every bounded rectangular derivation
Delta : A -> B(K,F) is completely bounded, with
||Delta||_cb <= 4 c_row ||Delta||.
Moreover some V in B(K,F) implements Delta:a |->V sigma(a)-lambda(a)V
with ||V|| <= c_0 ||Delta|| and c_0 = (1 + 4 c_row)^2.
Proof. Zero spaces are immediate, so assume all nonzero. Put
P = sigma(A)''. Fix h >= 1, X in M_h(A) with ||X|| <= 1, and
unit xi in K^h. Lemma 2.3 provides zeta in K and v in M_{h,1}(P)
with v zeta = xi and ||v|| <= 1.
Matrix Kaplansky density in the C*-algebra sigma(A) = A/ker sigma
gives a net of column contractions z_a in M_{h,1}(sigma(A))
converging strongly to v. Each z_a lifts to w_a in M_{h,1}(A)
with ||w_a|| <= 2, sigma_{h,1}(w_a) = z_a, and
sigma_{h,1}(w_a) zeta -> xi. (To lift, regard v as the first
column of a matrix in M_h(P) with all other columns zero;
approximate that matrix strongly by a net of contractions in
M_h(sigma(A)) via matrix Kaplansky density; compress back to the
first column; lift through the C*-quotient. The fixed slack
"2" accounts for the quotient norm slack.)
The entrywise derivation identity gives
Delta_h(X) sigma_{h,1}(w_a)
= Delta_{h,1}(X w_a) - lambda_h(X) Delta_{h,1}(w_a).
Taking norms, Lemma 2.1, ||w_a|| <= 2, ||X|| <= 1,
||lambda_h(X)|| <= 1:
|| Delta_h(X) sigma_{h,1}(w_a) zeta ||
<= c_row ||Delta|| ( ||X w_a|| + ||lambda_h(X)|| ||w_a|| )
<= 4 c_row ||Delta||.
The left factor Delta_h(X) is bounded at fixed finite h, so pass
to the limit sigma_{h,1}(w_a) zeta -> xi:
|| Delta_h(X) xi || <= 4 c_row ||Delta||.
Supremum over xi, X, h gives ||Delta||_cb <= 4 c_row ||Delta||.
Lemma 2.2 with the cb bound above yields the implementer with
||V|| <= d (1 + 4 c_row)^2 = c_0 ||Delta||. QED
2.6 The derivation criterion for similarity.
THEOREM 2.5 (Kirchberg 1996; per-algebra form in Ozawa 2006).
For a unital C*-algebra A, the following are equivalent:
(1) Every bounded unital homomorphism pi : A -> B(H), on
every Hilbert space H, is similar to a *-homomorphism.
(2) For every faithful unital *-representation sigma : A -> B(K),
every bounded derivation Delta : A -> B(K) is inner.
Consequence. It suffices to prove that every bounded derivation
into any faithful *-representation is inner. Parts 4-11 build the
machinery for that; Part 12 executes.
Note on Hypothesis (2). We do NOT require sigma faithful in the
proof, only that sigma be a given *-representation with a
faithful copy of A inside; the version with "every faithful
sigma" is the standard form and is what Theorem 2.5 provides.
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PART 3. THE UNIFORM COMMUTATOR ESTIMATE: REDUCTION PLAN
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THEOREM C restated. There is a finite absolute C with
||[Y^(h), X]|| <= C g_P(Y) ||X||
for every unital vN algebra P, every Y in B(K), h >= 1, X in M_h(P).
The constant we produce is
c_row = 4 sqrt(2), (Dickson)
c_0 = (1 + 4 c_row)^2, (cyclic-domain impl.)
c_1 = 6 pi^2 (1 + 4 c_0), (corner estimate)
C_fac = max { 3 c_1 + 2, 1 + 2 c_row, 2 }, (factor reps)
C = 3 C_fac + 2. (arbitrary vN algebra)
C is finite, explicit, universal. No optimality is claimed.
Numerically (c_row ~ 5.66): c_0 ~ 560, c_1 ~ 131,000,
C_fac ~ 393,000, C ~ 1,180,000.
The proof of Theorem C is a four-step refinement, all riding on the
SAME free-product corner estimate:
STEP 1 (Part 11): M a separable-predual II_1 factor in its
standard representation on L^2(M)^r for finite r.
||[Y^(h), X]|| <= (3 c_1 + 2) g_M(Y) ||X||.
STEP 2 (Part 12.1): P a unital factor on a separable Hilbert
space. Three cases: II_1, properly infinite, finite type I.
||[Y^(h), X]|| <= C_fac g_P(Y) ||X||.
STEP 3 (Part 12.2): P an arbitrary unital vN algebra on a
separable Hilbert space. Central direct integral decomposition.
||[Y^(h), X]|| <= C g_P(Y) ||X||.
STEP 4 (Part 12.3): arbitrary K. Reduce to a separable reducing
subspace containing all test vectors.
The engine of Step 1 is the following, proved in Parts 4-10:
THEOREM 3.1 (Free-product corner estimate).
Let L = D * A_0 be the tracial free product of a II_1 factor D
(generated by increasing unital full-matrix subalgebras D_j) and
a finite von Neumann algebra A_0 with a fixed faithful normal
tracial state. Let t = 1/N for an integer N >= 2, and let e be
a projection in D with tau(e) = t. Let G in B(L^2(L)) commute
with L_D. Suppose that for some epsilon >= 0,
||[G, L_x]|| <= epsilon ||x|| for all x in eLe.
Then for every s = s^* = s^{-1} in A_0 with tau(s) = 0,
||P_{L^2(A_0)} [G, L_s] |_{L^2(A_0)}|| <= c_1 epsilon,
c_1 = 6 pi^2 (1 + 4 c_0), c_0 = (1 + 4 c_row)^2.
The constant c_1 does NOT depend on t, N, G, s, or the ambient
space. That is the point: the corner estimate is what lets a
finitely-supported hypothesis become a universally-supported
conclusion without paying a price that grows as the corner shrinks.
Why Prove Theorem 3.1 with Such an Elaborate Machine.
The hypothesis is only that COMMUTATORS with corner elements are
small. The conclusion is a bound on the COMMUTATOR of G with L_s
COMPRESSED all the way down to L^2(A_0). Between these two lies an
enormous space of reduced words, and the argument must pass from
"small on a corner" to "small on L^2(A_0)" through a symmetry that
keeps a fixed A_0-vector intact. That passage needs a Fourier
inversion (a Hankel test) and a propagation through arbitrarily
long words. Parts 4-10 do that carefully; the constants accumulate
through specific, named estimates and no step multiplies them by
anything that depends on the ambient object.
Parts 4-6 prepare the structure of L^2(L) invariant under the
symmetry group of D. Parts 7-9 propagate the corner bound across
word levels. Part 10 performs the Hankel test and closes the
corner estimate.
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PART 4. FREE PRODUCTS, WORD GEOMETRY, TWO CLEANUPS
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4.1 The tracial free product.
Let D and A_0 be finite von Neumann algebras with fixed faithful
normal tracial states. Their FREE PRODUCT L = D * A_0 is the
unique tracial von Neumann algebra containing D and A_0 as
subalgebras with the same unit, such that the trace extends, and
such that the trace of any alternating product of centered
letters vanishes:
tau( x_1 x_2 ... x_n ) = 0
whenever every x_i has tau(x_i) = 0 and successive x_i lie in
different factors. (Here "centered" means tau(x_i) = 0.)
Equivalently, L is the von Neumann closure of the algebraic free
product in the tracial GNS representation.
Its L^2-completion L^2(L) is the Hilbert-space free product of
L^2(D) and L^2(A_0) relative to the vacuum. We will write down
exactly this decomposition.
4.2 Reduced words.
Put
E = L^2(D), E^o = E minus C 1,
B = L^2(A_0), B^o = B minus C 1.
For k >= 0 set
H_k = E tensor (E^o)^{tensor k} tensor J_k,
J_k = (B^o)^{tensor k} tensor B.
The LEVEL k SUBSPACE H_k is spanned by the tensors
d_0 tensor d_1 tensor ... tensor d_k tensor (a_1 tensor ... tensor a_k tensor z),
interpreted as the reduced word
d_0 a_1 d_1 a_2 ... a_k d_k z,
with d_0 in D unrestricted, z in A_0 unrestricted, d_1,...,d_k in
D^o = D minus C 1, a_1,...,a_k in A_0^o = A_0 minus C 1.
The scalar 1 of L corresponds to 1 tensor 1.
PROPOSITION 4.0 (Orthogonal word decomposition).
L^2(L) = orthogonal sum_{k >= 0} H_k.
Distinct tensor words are orthogonal: <w, w'> = 0 if w != w'.
The norm of a tensor word is the product of the L^2 norms of its
letters.
Proof. Combine freeness (tau of a reduced alternating word is zero
unless it is the scalar word) with the standard construction of
the Hilbert-space free product. The L^2 inner product <x, y> =
tau(y^* x) then makes the reduced words an orthogonal basis at
their indicated lengths. QED
Under this decomposition:
- L_D acts on the LEFT E factor only: L_d d_0 tensor ... = (dd_0) tensor ...
- L_{A_0} acts on the trailing B factor only.
- The relative commutant D' intersect B(L^2(L)) consists of
right multipliers R_b for b in D; the tracial commutation
theorem identifies it with {R_b}.
4.3 Cleanup 1: L is a factor.
LEMMA 4.1. D' intersect L = C 1. In particular L is a II_1 factor.
Proof. Regroup words so that the leftmost and rightmost letters in
the D-alphabet are exposed:
L^2(L) minus L^2(D) ~= E tensor V tensor E as D-D bimodules,
V = orthogonal sum_{n >= 1} B^o tensor (E^o tensor B^o)^{tensor (n-1)}.
Left D acts on the first E, right D on the last E. Expanding the
endpoints into C 1 + E^o recovers every reduced word containing at
least one A_0-letter exactly once.
Fix a Haar unitary v in D (which exists because D is a II_1 factor,
which contains a copy of L^infty(T)). Put u_n = v^n. Then u_n -> 0
ultraweakly in D: any normal functional restricts to
L^1(T) on W^*(v), whose Fourier coefficients tend to 0.
Consequently L_{u_n} -> 0 and R_{u_n^*} -> 0 weakly on E (test on
bounded vectors, then use L^2 density), and by uniform boundedness
both converge weakly to 0 on E tensor V tensor E, and the product
L_{u_n} R_{u_n^*} = L_{u_n} tensor 1 tensor R_{u_n^*} also tends
weakly to 0 there.
Now let z in D' intersect L be orthogonal to L^2(D). Then z is
orthogonal to every D-word and therefore lies in E tensor V tensor E
in the bimodule decomposition. D-centrality says
L_{u_n} R_{u_n^*} z = z for all n (since u_n (u_n)^* = 1). But
L_{u_n} R_{u_n^*} z -> 0 weakly, so z = 0. Hence z in L^2(D):
z = E_D(z) in D, and z commutes with D, so z in Z(D) = C 1 for the
II_1 factor D.
The center of L, Z(L), lies in D' intersect L. Every element of
Z(L) is fixed by every inner automorphism of D, so Z(L) is contained
in D' intersect L. By the first part, Z(L) = C 1. L has a faithful
finite trace (inherited) and diffuse subfactor D (which forces
infinite dimensionality), so L is a II_1 factor. QED
4.4 Cleanup 2: symmetrization over matrix stages.
The II_1 factor D is generated as a von Neumann algebra by an
increasing sequence D_1 subset D_2 subset ... of unital full-
matrix subalgebras, D_j ~ M_{m_j}(C) with m_j -> infinity.
For u in U(D_j) (the compact unitary group of the finite matrix
stage), let theta_u be the trace-preserving free-product
automorphism of L that equals Ad u on D and the identity on A_0.
Its Hilbert-space implementation W_u in U(L^2(L)) satisfies
W_u L_x W_u^* = L_{theta_u(x)}, W_u xi = xi for xi in L^2(A_0),
and W_u (d_0 tensor x_1 tensor ... tensor x_k tensor tail)
= theta_u(d_0) tensor theta_u(x_1) tensor ... tensor theta_u(x_k) tensor tail.
LEMMA 4.2 (Symmetrization). In proving Theorem 3.1, we may assume
in addition that G commutes with W_u for every u in the union
union_j U(D_j). The replacement preserves
||P_{L^2(A_0)} [. , L_s] |_{L^2(A_0)}||
exactly and preserves the corner hypothesis for both G and G^*.
Proof. For each j, average
G_j = integral_{U(D_j)} W_u G W_u^* dmu_j(u)
weakly, where mu_j is Haar probability. The unitary representation
u |-> W_u is strongly continuous on the compact group U(D_j);
averages in the weak operator topology are well defined and lie in
the norm ball of G.
Each conjugate W_u G W_u^* commutes with L_D and satisfies the
corner hypothesis with the same epsilon, because theta_u intertwines
D-corners of trace t. Averages preserve both. Pass to a weak
operator cluster point G~ as j -> infinity. Since the norm ball is
weakly compact, G~ exists; it commutes with L_D and satisfies the
corner hypothesis with the same epsilon. By Haar invariance, every
G_j with j >= i commutes with W_u for u in U(D_i); the limit G~
therefore commutes with every W_u for u in union_j U(D_j).
The compression P_{L^2(A_0)} [ - , L_s] |_{L^2(A_0)} is invariant
under u |-> W_u ( - ) W_u^*, because W_u fixes L^2(A_0) pointwise
and commutes with L_s. So the quantity we want to estimate is the
same for every conjugate, for every average, and for the limit.
Finally [G^*, L_x] = -[G, L_{x^*}]^*, so the corner hypothesis for
G gives it for G^*. QED
For the remainder of Parts 5-10, G denotes the symmetrized operator.
At the end we recover the compression estimate for the original
operator using Lemma 4.2.
4.5 What the extra symmetry buys.
Because G commutes with each W_u for u in U(D_j), a routine
approximation argument shows G commutes with the SIMULTANEOUS
adjoint action
C_u^{(tensor (k+1))} on E tensor (E^o)^{tensor k},
u in union_j U(D_j),
for every k >= 0. Lemma 4.2 lets us exploit this action via
finite-dimensional Schur-Weyl duality: on each stage D_j = M_m(C),
the operators C_u : x |-> u x u^* for u in U(D_j) generate the
full picture we need. Parts 5 and 6 exploit it.
================================================================
PART 5. INVARIANT TENSORS: THE CLASSIFICATION LEMMA
================================================================
We now exploit the symmetrization. The lemma of this Part is the
structural core of everything that follows: it says a bounded
multilinear map intertwining the left-D action and the simultaneous
conjugation action must be a "slot permutation", with no other
room.
Setup. Let k, l >= 0 and let
Y : E tensor (E^o)^{tensor k} -> E tensor (E^o)^{tensor l}
be bounded. Suppose
(I1) Y intertwines left D:
Y L_d^{(k)} = L_d^{(l)} Y for every d in D,
where L_d^{(k)} acts as L_d on the leftmost E and identity on
the remaining tensor factors;
(I2) Y intertwines the simultaneous conjugation C_u:
Y C_u^{(tensor (k+1))} = C_u^{(tensor (l+1))} Y
for every u in union_j U(D_j),
where C_u(x) = u x u^*.
Here E carries C_u, and E^o carries the same (it is C_u-invariant).
LEMMA 5.1 (Invariant intertwiners).
(1) If k != l, then Y = 0.
(2) If k = l, then there are scalars (a_sigma)_{sigma in S_k} with
Y = I_E tensor sum_{sigma in S_k} a_sigma U_sigma,
where U_sigma permutes the k interior tensor slots:
U_sigma(x_1 tensor ... tensor x_k)
= x_{sigma^{-1}(1)} tensor ... tensor x_{sigma^{-1}(k)}.
For k = 0 the sum reduces to a_() . I_E with a_() a single scalar.
Before proving, remark what makes this lemma hard. The two
hypotheses do not constrain the (k+1)-fold tensor E^{tensor(k+1)}
arbitrarily; they only pin it to the symmetric tensor-permutation
subspace on sufficiently large matrix stages. The proof must push
that finite-dimensional statement through the L^1 endpoint bound
that comes from traciality, then through a Hilbert-space dimension
argument to kill every would-be source-source or target-target
pairing.
Proof of Lemma 5.1.
Step 1 (endpoint bound from traciality). We assemble from Y a
multilinear form on unrestricted matrix entries. On D-matrices
a, x_1, ..., x_k, z_1, ..., z_l (no centering restricted on inputs,
centering applied inside), define
F(a, x, z) =
< Y(1 tensor x_1^o tensor ... tensor x_k^o),
a^* tensor (z_1^*)^o tensor ... tensor (z_l^*)^o >,
where x^o = x - tau(x) 1, and (z^*)^o = z^* - tau(z^*) 1.
Fix the tail vectors x_i, z_j and slice Y in those tail coordinates.
This yields an operator T : E -> E commuting with L_D, so by the
tracial commutation theorem T = R_b for some b in D, with
||b|| = ||T|| <= ||Y|| prod_i ||x_i^o||_2 prod_j ||(z_j^*)^o||_2.
Since F(a, x, z) = tau(a b), we obtain the ENDPOINT BOUND
|F(a, x, z)| <= ||Y|| ||a||_1 prod ||x_i||_2 prod ||z_j||_2. (5.1)
The slice also handles arbitrary bounded endpoints: replacing
1, a^* by d, c replaces a by c^* d (because <db, c> = tau(c^* d b)),
so the same bound holds with ||c^* d||_1 in place of ||a||_1.
Step 2 (finite matrix structure at each stage). Put p = 1 + k + l.
Restrict all p variables to a matrix stage D_j ~ M_m(C) for large
enough m. The intertwining hypothesis (I2) says F is invariant under
simultaneous conjugation u |-> u v u^*. Every multilinear form on
M_m^p has a unique expression as
Tr( A (v_0 tensor v_1 tensor ... tensor v_{p-1}) )
for A in End((C^m)^{tensor p}), using the nondegenerate trace
pairing. Conjugation invariance is equivalent to A commuting with
every u^{tensor p}, u in U(m). Finite-dimensional Schur-Weyl duality
says the commutant of U(m) on (C^m)^{tensor p} is the span of the
tensor-permutation operators P_pi (pi in S_p). So
F|_{D_j^p} = sum_{pi in S_p} lambda_{pi, j} Phi_pi,
Phi_pi(v_0, ..., v_{p-1})
= prod over cycles (i_1 ... i_r) of pi of tau(v_{i_1} ... v_{i_r}).
Uniqueness of this expansion on any stage of size m >= p, plus
consistency of the normalized trace under D_j subset D_{j'} (full
matrix inclusions preserve normalized trace), forces the
coefficients to stabilize: there are scalars lambda_pi with
F|_{D_j^p} = sum_pi lambda_pi Phi_pi for all j >= j_0.
Step 3 (endpoint bound kills short-cycle patterns). Fix pi in S_p.
Choose distinct matrix indices and set v_i = e_{i, pi(i)} for i =
0, 1, ..., p-1. Every nonzero product of these matrix units closes
into a cycle when traced; Phi_pi(v) = m^{-c(pi)} (where c(pi) is
the number of cycles of pi), and Phi_rho(v) = 0 for rho != pi.
Each matrix unit has ||v_i||_1 = m^{-1} and ||v_i||_2 = m^{-1/2}
(including diagonal units). By (5.1),
|lambda_pi| m^{-c(pi)} <= ||Y|| m^{-1 - (k+l)/2}.
Letting m -> infinity and remembering lambda_pi is fixed, we get
lambda_pi = 0 whenever c(pi) < 1 + (k+l)/2. (5.2)
Step 4 (pairing the tails). Restrict the k + l tail variables to
CENTERED matrices. Any form with a singleton tail cycle contributes
0 by centering. If there is no singleton tail, let r be the length
of the cycle of pi containing the distinguished variable a = v_0.
Its cycle count is at most
1 + (k + l - (r - 1))/2,
which is below the threshold (5.2) unless r = 1. If r = 1, the
threshold is attained only when every tail cycle has length
exactly 2. So a is a singleton and the remaining k + l tails pair
up. Consequently, on centered tails:
F(a, x, z) = tau(a) sum_{pairings P} lambda_P prod_{ {i,j} in P }
tau(v_i v_j), (5.3)
where v ranges over the combined list x_1, ..., x_k, z_1, ..., z_l.
Odd numbers of tails: no pairings, F = 0.
The formulas extend to arbitrary endpoints in E and arbitrary
centered L^2 tail letters. The union of matrix stages is L^2-dense
in E, the left side of (5.3) is continuous in its individual L^2
letters (by (5.1) and by continuity), and each right-hand factor is
continuous by |tau(uv)| <= ||u||_2 ||v||_2.
Step 5 (boundedness kills purely-internal pairings). Choose an
infinite orthonormal family (h_n) of self-adjoint centered vectors
in E (for example, the real trigonometric functions of a Haar
unitary in D, orthonormal by the group trace).
Fix a pairing P. Write r_s for the number of pairs entirely in
source slots x_1, ..., x_k, and r_t for the number in test slots
z_1, ..., z_l. Assign a disjoint infinite pool of h_n to each pair.
For each source-test pair, use one fixed h_n in its pool in both
slots. For each source-source pair, sum the source tensor over M
distinct members of its pool, filling both slots with the same
member. Same for test-test pairs independently.
Take both first endpoints (the inputs to the E-slot and its
conjugate) to be 1. In the expanded coefficient every label appears
exactly twice, in the positions assigned by P, and different pairs
use different labels. All other pairings yield zero. So the
resulting coefficient is lambda_P M^{r_s + r_t}. The source
tensor summands are orthonormal, as are the test ones, so the
product of the two tensor norms is M^{(r_s + r_t)/2}. Boundedness of
Y gives
|lambda_P| M^{r_s + r_t} <= ||Y|| M^{(r_s + r_t)/2}.
Let M -> infinity. Then lambda_P = 0 whenever r_s + r_t > 0.
The argument also covers the degenerate cases where one side has no
tail slots. So every surviving pair P joins a source slot to a
test slot. Such a pairing is exactly a bijection sigma in S_k
(the number of source slots equals the number of test slots), and
this requires k = l; if k != l, every lambda_P = 0, hence Y = 0.
Step 6 (identification). With the surviving pairings sigma in S_k,
(5.3) becomes
F(a, x, z) = tau(a) sum_{sigma in S_k} a_sigma prod_{i=1}^k
tau(x_i z_{sigma(i)}).
Using inner products <x, z> = tau(z^* x) for L^2 vectors, these are
the coefficients of the single tensor operator Y = I_E tensor
sum_sigma a_sigma U_sigma, tested against elementary tensors, and
density of elementary tensors identifies Y with that operator.
End of Lemma 5.1.
COROLLARY 5.2. If Y : E tensor (E^o)^{tensor k} -> E tensor
(E^o)^{tensor l} satisfies (I1)-(I2), then Y commutes with the
permutation that fixes a REPEATED letter. That is, if the same
centered vector x in E^o is inserted in every interior slot
(possibly equal to 0), then Y acts on tensors of the form
1 tensor x^{tensor k} tensor (tail) as
Y(1 tensor x^{tensor k} tensor tail)
= 1 tensor x^{tensor l} tensor (some operator on tail),
with the operator on tail independent of x.
Proof. By Lemma 5.1, Y is either 0 (k != l) or I_E tensor
sum_sigma a_sigma U_sigma (k = l); in either case it fixes x^{tensor k}
in the interior slot, since every U_sigma fixes x^{tensor k}. QED