An irrational angle theta on the unit circle X = R/Z defines a rotation
sigma of X and a pair of unitaries U, V on a Hilbert space H satisfying
VU = e^(2 pi i theta) UV. The von Neumann algebra M = W*(U, V) that U
and V generate is the hyperfinite II_1 factor R_theta. Its trace tau
is normalised by tau(1) = 1 and tau(U^k V^l) = 0 unless k = 0 = l.

For a continuous nonnegative f on X, set T_f = U f(V). The weighted
rotation acts on L^2(X, mu; l^2(Z)) by moving the k-th l^2 coordinate
of a vector at x to the (k+1)-st, times f(sigma^k(x)).

The Fuglede-Kadison determinant of T_f is exp of the integral of log f
over X, with log 0 = minus infinity. A continuous nonnegative f positive
almost everywhere whose integral of log f diverges to minus infinity
has determinant zero. That is the boundary case. The positive
determinant theory of Zhu, Fang, and Shi supplies many invariant
projections; zero determinant is open.
./pdf/pdf/fuglede-kadison-1952-determinant-theory.pdf

An invariant projection for T_f is a projection p in M with
0 < tau(p) < 1 and (1 - p) T_f p = 0. The stronger condition
pT_f = T_f p is not imposed. Zhu, Fang, and Shi, in a 2017 paper on a
class of operators in the hyperfinite II_1 factor, constructed such p
for strictly positive f with finite integral of log f, and posed the
complementary question for vanishing determinant.
./pdf/pdf/zhu-fang-shi-2017-hyperfinite-II1.pdf

Theorem. For every irrational theta in (0, 1) there exists a continuous
f: X -> [0, 1] with f^(-1)({0}) = {1} and integral log f divergent to
minus infinity, such that T_f = U f(V) has no invariant projection in
R_theta.

The weight is drawn after theta. No Diophantine arithmetic enters.

The Brown measure of T_f, in the sense of Dykema and Schultz's study of
Brown measure and iterates of the Aluthge transform, is the uniform
measure on the circle of radius the determinant, collapsing to the
Dirac mass at the origin when the determinant vanishes. A collapsing
Brown measure does not by itself block invariant projections. Dykema
and Haagerup constructed quasinilpotent DT-operators with many, and
Tucci constructed quasinilpotent generators of R_theta with non-trivial
affiliated invariant subspaces. The deciding information is the fine
structure of f.
./pdf/pdf/dykema-haagerup-2004-dt-operators.pdf

The theorem excludes invariant projections in M and nothing else. A
closed subspace of the ambient Hilbert space whose orthogonal
projection falls outside M is untouched. The positive-determinant
theorem of Zhu, Fang, and Shi shares this limitation.

Corollary. Every separable infinite-dimensional complex Hilbert space
carries a nonzero norm-quasinilpotent operator T whose commutant {T}'
is a proper, strongly closed, transitive unital algebra.

Proof. Fix an irrational theta and the weight f above, and let
T = T_f act on the tracial Hilbert space L^2(R_theta, tau). Quasinilpotence
is elementary. Put f = exp(-g) with g continuous, nonnegative, vanishing
only at 0, and integral g equal to plus infinity. Then T_f^n = U^n f[n](V)
with f[n] = product of f(sigma^k (.)), k in [0, n-1], and
||T_f^n|| = ||f[n]||_infty. Taking logarithms,
log ||T_f^n|| = sup_x sum_{k<n} log f(sigma^k(x)) = -inf_x sum_{k<n} g(sigma^k(x)).
Bare irrationality of theta makes sigma minimal, and g continuous,
so the average (1/n) sum_{k<n} g(sigma^k(x)) converges to integral g
uniformly in x as n grows. The sum diverges to plus infinity with n,
so log ||T_f^n|| -> minus infinity and ||T_f^n||^(1/n) -> 0.

Nonscalarity is immediate. The operator T_f is a weighted shift with
weight strictly less than one everywhere.

Let K be a closed hyperinvariant subspace of T_f. K is invariant under
every bounded operator commuting with T_f. The orthogonal projection p
of K commutes with the commutant {T_f}', because the commutant is a
star algebra and invariance under a star algebra makes the range, and
therefore the projection, reduce for the algebra. Thus p lies in
{T_f}''. Because T_f and its adjoint T_f^* = U^* f(V) both belong to M,
the von Neumann algebra {T_f}'' = W*(T_f, T_f^*, 1) sits inside M. So p
lies in M, and T_fp = pT_f. The invariance of K gives (1 - p) T_f p = 0.
The main theorem excludes such p, so K cannot be proper and nontrivial.

The commutant of a nonscalar bounded operator is a proper unital algebra
closed under strong limits. Properness: if {T}' = B(H), every rank-one
projection would commute with T, forcing every nonzero vector to be an
eigenvector; eigenvalues cannot vary on a linearly independent pair, and
the operator would be scalar, contradicting nonscalarity. Strong
closure: if A_n T = T A_n and A_n -> A strongly, both A_n T v and
T A_n v converge to A T v and T A v, so A T = T A. Transitivity of
{T}' is exactly the absence of nontrivial hyperinvariant subspaces of T,
since a hyperinvariant subspace is a common invariant subspace of the
commutant.

Cho, Ko, and Lee, in a recent preprint, claimed at parameter mu = 1
that a norm-quasinilpotent operator with certain auxiliary properties
has a nontrivial hyperinvariant subspace. The hypotheses apply to
I + T_f, whose commutant coincides with {T_f}', and T_f has no such
subspace. Only the hyperinvariant-subspace claim at mu = 1 is
contradicted by the present construction; the rest of the Cho-Ko-Lee
theorem is untouched.

The construction separates an obstruction from its realization as a
continuous weight. An inequality of convolution type is established for
the measurable weight under rich families of ordered pairs (x, y)
drawn from a mixed digit expansion of the circle, and after that the
measurable weight is smoothed into a continuous weight with a single
zero, preserving the obstruction by an affiliated multiplier. No step
uses more than bare irrationality of theta.