Hyperinvariant subspaces and commutants

For a bounded operator \(T\) on a complex Hilbert space \(\mathcal H\), write \(\{T\}'=\{A\in B(\mathcal H):AT=TA\}\). A closed subspace is hyperinvariant for \(T\) if every member of \(\{T\}'\) leaves it invariant. An operator algebra is transitive if its only common closed invariant subspaces are \(0\) and \(\mathcal H\).

The bridge to the containing von Neumann algebra is the elementary observation recorded in Dykema (2005, Proposition 2.1).

These conclusions do not exclude ordinary invariant subspaces in the representation. A separate companion gives a direct \(2\)-adic weighted shift with a transitive commutant, including an explicit norm estimate, backward intertwiners and a description of its polar conjugates (OpenAI 2026). It uses only Lemmas 2 and 9 from this paper. The present proof uses no result from that companion.

Prior work

The determinant of Fuglede and Kadison (Fuglede and Kadison 1952) and Brown’s spectral distribution (Brown 1986) form a major route to invariant subspaces in finite factors. Haagerup and Schultz associate an affiliated hyperinvariant projection to each Borel subset of the plane, with trace equal to its Brown mass (Haagerup and Schultz 2009, Theorem 1.1). Thus a Brown measure that is not a point mass gives a nontrivial projection. For a point-mass Brown measure, the trace formula gives only trivial projections.

Point-mass Brown measure can nevertheless coexist with many invariant subspaces. Dykema and Haagerup construct a one-parameter family of nontrivial affiliated hyperinvariant subspaces for the quasinilpotent DT-operator (Dykema and Haagerup 2004, Theorems 2.2 and 2.9). Thus degeneration of Brown measure to a point mass does not itself exclude invariant projections. Tucci also constructed quasinilpotent generators of the hyperfinite factor with nontrivial affiliated invariant projections (Tucci 2008, Theorems 3.1 and 3.3 and the discussion following Theorem 3.3). These examples further distinguish quasinilpotence from the absence of factor-invariant projections.

For weighted irrational rotations, the measurable-action theorem of Dykema and Schultz identifies the Brown measure as uniform measure on the circle whose radius is the determinant (Dykema and Schultz 2009, Theorem 5.4). Zhu, Fang, and Shi give a direct proof, explicitly crediting that earlier result (Zhu et al. 2017, Theorem 5.3). At determinant zero this measure is the point mass at zero, so the Brown-measure projections are trivial.

Continuous weights with one zero and determinant zero were already exhibited by Zhu, Fang, and Shi (Zhu et al. 2017, 262–63). The additional conclusion of Theorem 1 is the exclusion of every nontrivial invariant projection for a suitable weight at each prescribed angle. Zhu, Fang, and Shi’s spectral-radius formula (Zhu et al. 2017, Theorem 2.3) implies quasinilpotence in this case. We give an elementary proof of this special case, using only uniform averaging of continuous functions under irrational rotation. Their positive-determinant invariant-subspace result (Zhu et al. 2017, Corollary 5.5) has a hypothesis complementary to the one studied here.

Proof strategy

The obstruction begins with a measurable weight. Suppose its weighted rotation had a nontrivial invariant projection. In the shift representation, this projection becomes a field of projections. The finite trace and ergodicity force its central diagonal entry to lie strictly between zero and one almost everywhere. Thus both the projection and its complement have vectors with a nonzero central coordinate.

Invariance imposes weighted orthogonality identities on these vectors. We first obtain the identities along rotation orbits and then pass them to suitably coupled pairs of points. The weight has drops that are rare enough to leave a chosen finite window under control, but strong enough to suppress contributions outside that window. The central coordinates force at least one finite convolution coefficient to stay away from zero. The suppressed tail cannot cancel it, contradicting the exact orthogonality identity. Conditioning separately on each endpoint controls that tail even though it has infinitely many terms.

The measurable weight is then replaced by a continuous weight with one zero. Smoothing and moving the logarithmic drops changes their sum by a constant plus a measurable coboundary. This relation supplies an intertwining multiplier, possibly unbounded. A domain and trace argument shows that an invariant projection for the continuous operator would produce one for the measurable operator with exactly the same trace. This transfers the obstruction to the desired continuous weight.

The construction realizing this scheme is as follows. Use the additive circle \(X=\mathbb R/\mathbb Z\), with Haar probability \(\mu\), and write \(\sigma x=x+\theta\). Write \(\{t\}\in[0,1)\) for fractional part. We construct divisible integers \(1=q_0\mid q_1\mid q_2\mid\cdots\) and positive costs \(c_i\). The measurable model is the weight \[g_*(x)=\sum_{i\ge1}c_i\lfloor\{q_ix\}+\{q_i\theta\}\rfloor, \qquad f_*(x)=e^{-g_*(x)}.\] The series is finite almost everywhere although its integral diverges. Large divisible frequencies allow its trial phases to be realized at the fixed angle \(\theta\), while finitely many sampling tests approximate independent Haar coordinates. The circle characters themselves are dependent. A mixed-radix expansion supplies independent digits, from which we build the ordered couplings with controlled marginal densities. The prescribed-marginal viewpoint belongs to the classical coupling theory of Strassen and Edwards (Strassen 1965; Edwards 1978); here we also need approximation by positive orbit pairs.

Section 2 fixes the frequencies and sampling powers. Sections 3 and 4 obtain the projection identities on the ordered pairs. Section 5 proves and applies the convolution obstruction, and Section 6 constructs the continuous weight and proves the trace-preserving transfer.

Appendix 7 gives a complete alternative realization on an infinite product of circles. Independent coordinates yield a finite-coordinate orbit-sampling theorem and a second application of the same convolution obstruction. That model also separates the atomic algebras generated by finitely many polar conjugates from the diffuse algebra generated by the full orbit. The prescribed-angle circle proof is complete before the appendix.

All identities involving measurable fields hold on appropriate conull sets. When integer iterates are used, we intersect the countably many translates of these sets. The constants implicit in \(O(\cdot)\) below are absolute unless a fixed construction stage is specified. Every new ingredient needed for the proof is established in the paper.

Fields determined by scalar multipliers

We first isolate the elementary field representation used here and in the direct weighted-shift construction. It depends only on commutation with scalar multipliers.

The tracial representation

We use the standard crossed-product representation for a probability-preserving action; see Anantharaman and Popa (n.d., sec. 1.4 and Proposition 1.4.5). We verify below the field and trace properties needed in the proof.

Write \(X=\mathbb R/\mathbb Z\), let \(\mu\) be Haar probability measure, and put \(\sigma x=x+\theta\). On \(\mathcal H=L^2(X,\mu;\ell^2(\mathbb Z))\), let \(S e_k=e_{k+1}\) and define \[U(x)=S,\qquad \pi(a)(x)e_k=a(\sigma^k x)e_k\quad(a\in L^\infty(X,\mu)).\] For \(V=\pi(x\mapsto e^{2\pi i x})\), these operators satisfy \(VU=e^{2\pi i\theta}UV\). We also write \(e_0\) for the constant field with this value. The vector \(U^mV^n e_0\) is the field \(x\mapsto e^{2\pi i nx}e_m\). These vectors span a dense subspace by ordinary Fourier theory, and \(\langle U^mV^n e_0,e_0\rangle\) equals \(1\) for \((m,n)=(0,0)\) and \(0\) otherwise. The commutation relation reduces every product of monomials to a scalar multiple of a monomial, so these moments identify all inner products of polynomial vectors with the prescribed tracial cyclic representation. The resulting unitary identifies the generated algebra \(M\) with \(R_\theta\) and its faithful normal trace with \(\tau\). Measurable functional calculus for \(V\) places every \(\pi(a)\) in \(M\).

Every generator commutes with scalar multiplication by a function of \(x\). Lemma 9 therefore represents every \(A\in M\) by an essentially bounded measurable field. The generators also commute with the unitary \(J\) given by \((J\zeta)(x)=S\zeta(\sigma x)\): for \(\pi(a)\) the coordinate calculation is \(a(\sigma^{k-1}(\sigma x))=a(\sigma^kx)\). These commutations pass to their von Neumann algebra. Testing \(AJ=JA\) on the constant basis fields gives \[ A(\sigma x)=S^{-1}A(x)S\quad\text{for almost every }x. \tag{15}\] The vector formula for the trace is now \[ \tau(A)=\int_X\langle A(x)e_0,e_0\rangle\,d\mu(x) \qquad(A\in M). \tag{16}\] We choose Borel versions for projection fields and discard null sets and their countably many rotation translates when necessary. Irrational rotation is ergodic: all Fourier coefficients of nonzero index vanish for an invariant \(L^2\) function. We use equidistribution in the form established at the beginning of Section 2.

The following consequence of covariance will also apply to the product model. It ensures that a nontrivial invariant projection has a nonzero zeroth coordinate at almost every endpoint, as does its complement.

A compact-set transfer lemma

Projection fields need not be continuous. The following observation allows us to pass an identity to a weak limit of endpoint measures, provided that their marginals do not concentrate on small sets. A family of finite measures is uniformly absolutely continuous with respect to \(\mu\) if, for each \(\varepsilon>0\), one can choose \(\delta>0\) such that every set of \(\mu\)-measure below \(\delta\) has measure below \(\varepsilon\) for every member of the family.

The measurable weight and its cocycles

We return to the prescribed rotation on \(X=\mathbb R/\mathbb Z\). The next formulas supply its positive weight and the diagonal ratios to which Lemma 11 will apply. For the parameters of Proposition 8, put \[ g_*(x)=\sum_j C_j\sum_{i\in J_j} \lfloor u_i(x)+\alpha_i\rfloor,\qquad f_*(x)=e^{-g_*(x)},\qquad T_*=U\pi(f_*). \tag{18}\] The sum defining \(g_*\) has only finitely many nonzero terms almost everywhere. Here \(c_i=C_j\) for \(i\in J_j\) in the introduction’s notation. To prove the finiteness assertion, observe that \[\sum_i\mu\{\lfloor u_i+\alpha_i\rfloor\ne0\} =\sum_i\alpha_i\le2\sum_j K_j^{-1}<\infty.\] The elementary first Borel–Cantelli implication follows here simply by bounding the measure of a union of tail events by the sum of their measures. Thus \(0<f_*\le1\) almost everywhere.

For each \(k\in\mathbb Z\), define \[ F_k(x)=\exp\left(-\sum_j C_j\sum_{i\in J_j} \lfloor u_i(x)+k\alpha_i\rfloor\right). \tag{19}\] These sums, too, have only finitely many nonzero terms almost everywhere: in the tail where \(|k|\alpha_i<1\), the probability of a nonzero floor is \(|k|\alpha_i\). After taking a countable intersection, there is an invariant conull set on which all \(F_k\) are finite and strictly positive and \[ F_{v+n}(x)=F_v(x)F_n(\sigma^v x),\qquad F_0=1,\qquad F_1=f_*. \tag{20}\] Indeed \(u_i(\sigma^v x)=\{u_i(x)+v\alpha_i\}\), so the addition formula for floors proves the identity term by term. Since \(\alpha_i>0\), \(F_k(x)\) is nonincreasing in \(k\). Give these functions arbitrary positive finite Borel values on the remaining null set.

For endpoints in the common conull set, write \[R_{x,y}(k)=F_k(y)/F_k(x).\] If \(u_i(x)\le u_i(y)\), the \(i\)th coordinate contributes at most \(1\) to this ratio. For \(i\in J_s\), it contributes \(e^{-C_s}\) precisely when \[ u_i(x)<1-\{k\alpha_i\}\le u_i(y). \tag{21}\] When \(k\alpha_i\) is integral the displayed event is empty, as it should be. Without an order assumption, an individual coordinate contributes at most \(e^{C_s}\), since the two floors differ by at most one.

From invariant projections to endpoint identities

Suppose \(p\in M\) is a projection satisfying \((1-p)T_*p=0\). Its range is then invariant under every positive power of \(T_*\). The formula \[T_*(x)^n=S^n\operatorname{diag}_v F_n(\sigma^v x)\] and covariance (15) show that, for \(y=\sigma^n x\), \[ (1-p(y))\operatorname{diag}_v R_{x,y}(v)p(x)=0\qquad(n\ge1). \tag{22}\] To verify the division involved, the cocycle identity gives \(F_n(\sigma^v x)/F_n(x)=F_v(\sigma^n x)/F_v(x)\); the positive divisor \(F_n(x)\) is finite. The resulting diagonal is bounded on each such fiber by \(1/F_n(x)\), since \(F_n\le1\). This bound need not be uniform in \(x\).

If \(0<\tau(p)<1\), then \[ 0<r(x):=\langle p(x)e_0,e_0\rangle<1 \quad\text{for almost every }x. \tag{23}\] This is Lemma 10, applied with \(f=f_*\), the faithful trace (16), and covariance (15).

The next lemma is the payoff from the sampling construction. It extends (22) beyond actual orbit pairs, while retaining a bound on every diagonal entry at once.

A conditional convolution estimate

We isolate the common argument so that the product construction can use it without repeating the infinite-tail estimate. The probability space in the lemma may include auxiliary labels besides its endpoints.

Application to the circle weight

Suppose, for contradiction, that such a projection exists. Put \[\xi(x)=p(x)e_0,\qquad \gamma(y)=(1-p(y))e_0, \qquad r(x)=p(x)_{0,0}.\] Both vector norms are at most \(1\). We shall use the laws \(\mathbb P_j\) of Section 4; their expectations are denoted by \(\mathbb E_j\).

The exact convolution identities

By (28), the vector \(\operatorname{diag}_v(R_{x,y}(v))\xi(x)\) belongs to \(\operatorname{ran}p(y)\). Applying \(T_*(y)^n\), which preserves that range for \(n\ge0\), and pairing with \(\gamma(y)\) gives \[ 0=\sum_{v\in\mathbb Z}\frac{F_{n+v}(y)}{F_v(x)} \xi_v(x)\overline{\gamma_{n+v}(y)} =\sum_{v\in\mathbb Z} A_vB_{-n-v},\qquad n\ge0, \tag{35}\] where, in this section, \[A_v=\frac{\xi_v(x)}{F_v(x)},\qquad B_v=F_{-v}(y)\overline{\gamma_{-v}(y)}.\] All these identities hold simultaneously almost surely under each \(\mathbb P_j\). They are absolutely convergent. Indeed, monotonicity of \(F_k\) in \(k\) and coordinate order imply \[ \frac{F_{n+v}(y)}{F_v(x)} \le\min\{R_{x,y}(v),R_{x,y}(n+v)\}\le1, \tag{36}\] and \(\sum_v|\xi_v(x)\gamma_{n+v}(y)|\le1\) by Cauchy–Schwarz. The comparison with \(R_{x,y}(v)\) follows from \(F_{n+v}(y)\le F_v(y)\); the comparison with \(R_{x,y}(n+v)\) follows from \(F_{n+v}(x)\le F_v(x)\).

A finite convolution cannot be too small

Fix \(j\) and abbreviate \(K=K_j\), \(m=m_j\). Define an event \(E_j\) by the following three conditions: \[\begin{align*} &X_i<1-3/m,\quad Y_i>3/m\quad(i\in J_j), \quad\text{and at least one tag in }J_j;\\ &X_i<1-3K/N_s,\quad Y_i>3K/N_s \quad(i\in J_s,\ s>j);\\ &r(x)\ge1/j,\qquad 1-r(y)\ge1/j. \end{align*}\] For \(0\le n\le2K\), let \[ c_{j,n}=\sum_{\substack{|v|\le K\\ |n+v|\le K}}A_vB_{-n-v}. \tag{37}\]

Controlling every part of the infinite tail

To contradict (38), we must bound the omitted terms of (35). The important estimate is conditional on one endpoint. It will let us sum infinitely many terms without taking a union bound over their exceptional events.

Completion of the contradiction

Apply Lemma 17 to the labelled law \(\mathbb P_j\), with \[K=K_j,\quad D=D_j,\quad a=1/j,\quad c=c_* =e^{-90}/2, \qquad \eta=e^{-L_j},\quad \delta=e^{-L_j/400}.\] Equations (35) and (36) verify its algebraic assumptions. The proof of Lemma 19 supplies the event of mass at least \(c_*\) and all central-coefficient bounds for sufficiently large \(j\). Lemma 20 gives the tail estimate with conditional probabilities under the full law. Consequently \[c_*\varepsilon_j \le(2K_j+1)\bigl(e^{-L_j}+2e^{-L_j/800}\bigr) \le\frac{\varepsilon_j}{j},\] where the last inequality is the parameter choice (10). Since \(\varepsilon_j>0\) and eventually \(1/j<c_*\), this is impossible. This proves Theorem 18.

Transport by an affiliated positive operator

A positive self-adjoint operator is affiliated with a von Neumann algebra if all its spectral projections belong to that algebra. Such an operator commutes, including on its domain, with every unitary in the commutant. The action of an invertible affiliated operator on projection lattices also appears in Mori (2020, Proposition 3.1). The form needed here includes a specified intertwining domain and preservation of the trace; we give a direct spectral-cutoff proof.

Smoothing and moving the logarithmic drops

We claim that there exist a nonnegative function \(g\), finite and continuous on \(X\setminus\{0\}\) and tending to infinity at \(0\), and an almost everywhere finite real measurable function \(b\) such that \[ g=g_*+1+b\circ\sigma-b\quad\text{almost everywhere}, \qquad \int_Xg\,d\mu=\infty. \tag{42}\] The construction has three parts: smooth each drop, move its support into a shrinking arc, and add an integrable background cost that guarantees divergence at the center of those arcs. Apart from the constant \(1\), all changes will be explicit coboundaries.

Fix \(i\in J_j\) and put \(\delta_i=\alpha_i/4\). For \(u\in X\), define the continuous periodic smoothing of the sawtooth by \[s_i(u)=\frac1{2\delta_i}\int_{-\delta_i}^{\delta_i}\{u+t\}\,dt.\] It equals \(\{u\}\) outside the \(\delta_i\)-neighborhood of the cut at \(0\). Set \[h_i(x)=C_j\bigl(\alpha_i+s_i(u_i(x)) -s_i(u_i(x)+\alpha_i)\bigr).\] The elementary identity \(\lfloor\{u\}+\alpha_i\rfloor= \alpha_i+\{u\}-\{u+\alpha_i\}\) shows that \(h_i\) is the corresponding average of the original one-coordinate cost. It is nonnegative and continuous, with \[ \int_X h_i\,d\mu=C_j\alpha_i, \qquad \mu(\operatorname{supp}h_i)\le\alpha_i+2\delta_i. \tag{43}\] For \[d_i^{\rm sm}(x)=C_j\bigl(s_i(u_i(x))-u_i(x)\bigr),\] we have the exact measurable identity \[ h_i(x)=C_j\lfloor u_i(x)+\alpha_i\rfloor +d_i^{\rm sm}(x)-d_i^{\rm sm}(\sigma x). \tag{44}\] The support of \(d_i^{\rm sm}\) has measure at most \(2\delta_i\).

Recall the shrinking arcs \(O_j\), the integers \(h_j^*\), and the continuous partition of unity \(\phi_{j,n}\), \(0\le n\le h_j^*\), prepared in the parameter construction, with \(\sigma^n(\operatorname{supp}\phi_{j,n})\subset O_j\). Define \[h'_i(x)=\sum_{n=0}^{h_j^*}(h_i\phi_{j,n})(\sigma^{-n}x).\] This is continuous and nonnegative, supported in \(O_j\), and has the same integral as \(h_i\). Telescoping gives \[ h'_i=h_i+d'_i-d'_i\circ\sigma, \qquad d'_i(x)=\sum_{n=1}^{h_j^*}\sum_{r=1}^n (h_i\phi_{j,n})(\sigma^{-r}x). \tag{45}\] Although this transfer contains a double sum, its support is contained in just \(h_j^*\) translates of \(\operatorname{supp}h_i\), one for each possible \(r\). Consequently \[\sum_i\left(\mu\{d_i^{\rm sm}\ne0\}+\mu\{d'_i\ne0\}\right) \le C\sum_j\frac{1+h_j^*}{K_j}<\infty.\] Thus each of the transfer sums has only finitely many nonzero terms almost everywhere, including at the translated point \(\sigma x\).

The sum \(\sum_i h'_i\) is locally finite on \(X\setminus\{0\}\) because its successive blocks are supported in \(O_j\downarrow\{0\}\). It is therefore continuous there. Its integral is infinite: Tonelli’s theorem and \(\alpha_i>1/(m_jK_j)\) give \[ \int_X\sum_i h'_i\,d\mu =\sum_j C_j\sum_{i\in J_j}\alpha_i \ge\sum_j C_j/K_j=\infty, \tag{46}\] since \(C_j\ge K_j^2\). Local finiteness alone need not force divergence at \(0\); the following background supplies that remaining property.

A background cost diverging at the zero

Let \(W_j=2^{j+1}\). Choose continuous functions \(0\le v_j\le1\) with \(v_j(0)=1\), supported in closed arcs \(I_j\) shrinking to \(\{0\}\). Choose these arcs small enough that \(\sigma I_j,\ldots,\sigma^{W_j}I_j\) are pairwise disjoint and \[\sum_j W_j\mu(I_j)<\infty.\] This is possible because each finite set of distinct points \(\theta,2\theta,\ldots,W_j\theta\) has positive minimum separation. For \(x\ne0\), put \[g_0(x)=1+\sum_j\left(v_j(x)-\frac1{W_j} \sum_{n=1}^{W_j}v_j(\sigma^{-n}x)\right).\] For each \(j\), disjointness bounds the norm of the subtracted function by \(1/W_j\). These bounds sum to \(1/2\), so the negative series converges uniformly. The positive series is locally finite away from \(0\). Consequently \(g_0\) is continuous there and \(g_0\ge1/2\). For every fixed integer \(J\), the finite sum \(\sum_{j\le J}v_j(x)\) tends to \(J\) as \(x\to0\). The bounded negative series therefore implies \(g_0(x)\to\infty\) as \(x\to0\). Its integral is finite, since \(\sum_j\int v_j\,d\mu\le\sum_j\mu(I_j)<\infty\) and the negative series is uniformly bounded.

Define \[b_0(x)=\sum_j\frac1{W_j}\sum_{n=1}^{W_j}\sum_{r=1}^n v_j(\sigma^{-r}x).\] The \(j\)th summand is supported in \(\bigcup_{r=1}^{W_j}\sigma^r I_j\). The assumed summability makes this sum finite almost everywhere. On a conull set where the sums at \(x\) and \(\sigma x\) are finite, telescoping shows \[g_0=1+b_0\circ\sigma-b_0.\] Now let \[g(x)=g_0(x)+\sum_i h'_i(x)\quad(x\ne0),\qquad g(0)=\infty, \qquad b=b_0-\sum_i(d_i^{\rm sm}+d'_i).\] The preceding arguments prove that \(g\) is nonnegative, finite and continuous off \(0\), tends to infinity at \(0\), and has infinite integral. The transfer \(b\) is real and finite almost everywhere. Summing (44) and (45) is legitimate on the common conull set where the transfer sums and the original cost have finitely many nonzero terms. Together with the background identity this proves (42).

Conclusion and spectral consequences

Define the function on the unit circle by \[f(e^{2\pi i x})=e^{-g(x)},\qquad e^{-\infty}=0.\] It is continuous, is positive away from \(1\), and vanishes at \(1\). Its modulus is nonconstant and its zero set has Haar measure zero. Moreover, \[ \int_{\mathbb T}\log f\,dm=-\infty. \tag{47}\] In particular, its Fuglede–Kadison determinant is zero (Fuglede and Kadison 1952): the spectral distribution of \(\pi(f)\) is the Haar distribution of \(f\), so the determinant definition is precisely the exponential of (47). Also \(|U\pi(f)|=\pi(f)\), which gives the same determinant for \(T_f=U\pi(f)\). This conclusion uses the nonnegative integral calculation (46); it does not require the transfer \(b\) to be integrable.

Choose a finite Borel version of \(b\) and discard all integer translates of the null set where the coboundary identity fails. Let \(H_b\) be multiplication on the \(k\)th fiber coordinate by \(e^{b(\sigma^k x)}\), with its natural square-integrability domain. It is positive, self-adjoint, and injective. Its spectral projections are \(\pi\)-images of spectral indicators of \(e^b\), hence lie in \(M\). Thus \(H_b\) is affiliated with \(M\). The coboundary identity gives the pointwise factor equality \[e^{b(\sigma^{k+1}x)}e^{-g(\sigma^k x)} =e^{-1}f_*(\sigma^k x)e^{b(\sigma^k x)}.\] For every \(\zeta\in\operatorname{Dom}H_b\), the right-hand side defines the \(k+1\) coordinate of \(e^{-1}T_*H_b\zeta\), a square-integrable vector because \(T_*\) is bounded. Therefore \[T_f\zeta\in\operatorname{Dom}H_b, \qquad H_bT_f\zeta=e^{-1}T_*H_b\zeta.\] Lemma 21 sends any invariant projection of \(T_f\) to an invariant projection of \(e^{-1}T_*\) having the same trace. Multiplication by the nonzero scalar \(e^{-1}\) does not change invariant ranges. Theorem 18 therefore excludes traces strictly between \(0\) and \(1\). This proves Theorem 1 for the arbitrary irrational \(\theta\) fixed at the start.

The same continuous weight has a stronger spectral property, which we record with an elementary proof. This is the zero-determinant case of Zhu et al. (2017, Theorem 2.3). We include the truncated-logarithm argument, which uses uniform averaging under irrational rotation.

Scales and independent angles

We choose the finite polynomial accuracy before choosing each new block. This leaves enough freedom to obtain both the sampling estimates and the threshold crossings needed by Lemma 17. All choices in this subsection are independent of any invariant projection.

Partition \(\mathbb N=\{1,2,\ldots\}\) into consecutive finite blocks \(J_j\) of length \(m_j\), writing \[d_0=0,\qquad d_j=\sum_{s\le j}m_s,\qquad J_j=\{d_{j-1}+1,\ldots,d_j\}.\] The cutoff \(K_j\) will specify the finite coefficient window in the contradiction, while \(D_j\) and \(\varepsilon_j\) will give the coefficient bound and the product lower bound from Lemma 7. After these are fixed, \(L_j\) specifies the required decay of the omitted terms; the logarithmic drop \(C_j\) and the block length \(m_j\) are chosen to achieve that decay while retaining uniform bounds on the coupling marginals.

We now choose these parameters. Put \(\kappa_j=1/(100j)\) and start with \(K_1=100\) and \(G_1=1\). At stage \(j\), the numbers \(K_j,G_j\) and all parameters of earlier blocks have already been fixed. Make the following choices in the stated order.

1. Define \[D_j=\exp\left((K_j+1)\sum_{s<j}C_sm_s\right).\] Apply Lemma 7 with \(K=K_j\), \(D=D_j\), and \(\delta=1/j\) to choose \(\varepsilon_j>0\). This is possible because \(D_j\ge1\ge1/j\). Consequently, \[ \max_{0\le n\le2K_j}|(AB)_{-n}|\ge\varepsilon_j \tag{48}\] whenever \(A,B\) are supported in \([-K_j,K_j]\), their coefficients have modulus at most \(D_j\), and \(|A_0|,|B_0|\ge1/j\).

2. Choose \(L_j\ge1\), increasing with \(j\), so large that \[ (2K_j+1)3\exp(-L_j/10000)\le\frac{\varepsilon_j}{j}. \tag{49}\] Set \(C_j=\max\{L_j,K_j^2\}\), and choose a finite \(G_{j+1}\ge1\) such that \[ \kappa_{j+1}\sqrt{G_{j+1}}\ge L_j. \tag{50}\]

3. Choose an integer \(m=m_j\) sufficiently large, and set \[K_{j+1}=\left\lceil m^{(3/4)m}\right\rceil,\] subject to \[ \begin{gathered} m\ge\max\{K_j,2^j,4G_j,100\},\qquad d_{j-1}\le m/100,\\ \kappa_jm^{1/40}\ge L_j,\qquad G_{j+1}K_{j+1}<m^{(4/5)m}. \end{gathered} \tag{51}\]

Every choice is finite and feasible. Indeed, the first two steps do not depend on the new \(m_j\). All but the final requirement in (51) are lower bounds on \(m\). For the final one, the already fixed number \(G_{j+1}\) satisfies \[\frac{G_{j+1}\lceil m^{(3/4)m}\rceil}{m^{(4/5)m}} \le 2G_{j+1}m^{-m/20}\longrightarrow0.\] There is therefore no upper restriction conflicting with the lower bounds.

Write \[ N_j=m_jK_j,\qquad a_j=G_j/m_j,\qquad b_j=m_j^{-19/20}. \tag{52}\] The construction gives the useful consequences \[ \begin{gathered} m_j\ge2^j,\qquad d_j\le1.01m_j,\qquad N_j\le m_j^2,\\ K_{j+1}>2K_j,\qquad \sum_{s>j}\frac1{K_s}\le\frac2{K_{j+1}},\qquad \frac1{m_j}\le a_j,b_j\le\frac14. \end{gathered} \tag{53}\] For example, \(m_j\ge100\) and \(m_j\ge K_j\) imply \(K_{j+1}\ge m_j^{(3/4)m_j}>2m_j\ge2K_j\), and the reciprocal estimate follows by summing a geometric majorant. Also \(m_{j+1}\ge K_{j+1}>m_j\), so each of \(m_j,K_j,N_j\) is increasing. The upper bound for \(a_j\) follows from \(m_j\ge4G_j\); the other width bounds follow from \(G_j\ge1\) and \(m_j\ge100\).

Use the coordinate order on \(X\) defined above.

On a separate product probability space choose independent random angles \(\theta_i\), with \(\theta_i\) uniform on \([1/N_j,2/N_j]\) when \(i\in J_j\), and let \[\sigma_\theta x=x+\theta\quad\text{in }X.\] For every realization, \[ \sum_{i=1}^\infty\theta_i \le2\sum_{j=1}^\infty\frac{m_j}{N_j} =2\sum_{j=1}^\infty\frac1{K_j}<\infty. \tag{54}\] Almost surely \(1,\theta_1,\theta_2,\ldots\) are rationally independent. To see this, multiply any proposed finite rational relation by a common denominator. If some angle has a nonzero integer coefficient, conditioning on all other angles leaves one prescribed value of an atomless random variable, an event of probability zero. A relation with no angle term cannot hold unless its constant term is zero. There are only countably many nontrivial finite integer relations.

For a rationally independent realization the rotation is ergodic and free. Indeed, the characters \[x\longmapsto\exp\left(2\pi i\sum_i v_ix_i\right), \qquad v_i\in\mathbb Z,\quad v_i=0\text{ for all but finitely many }i,\] form an orthonormal basis of \(L^2(\mu)\): finite-coordinate functions are dense, and the usual Fourier bases are complete on each finite torus. Translation multiplies this character by \(\exp(2\pi i\sum_i v_i\theta_i)\), which equals \(1\) only for the constant character. Hence the invariant \(L^2\) functions are constant, which is ergodicity. Also \(n\theta_1\notin\mathbb Z\) for every \(n\in\mathbb Z\setminus\{0\}\), so \(\sigma_\theta^n x\ne x\) for every \(x\in X\).

We retain the random angles until the sampling construction below has supplied one further probability-one property.

Sampling ordered orbit pairs

We next approximate prescribed changes in finitely many coordinates by positive orbit moves. Both endpoint distributions must remain controlled: this will allow the projection identity to pass from orbit pairs to the couplings used in the product construction. The scalar estimates of Lemma 6 apply coordinate by coordinate; we prove here the growing-product estimate and the ordered weak limit.

The coordinate order is only Borel in the product-circle topology: already in one circle, the ordered pairs \((1/2,1-1/n)\) converge to the unordered pair \((1/2,0)\). Thus the closed-relation hypothesis in the classical coupling theorem (Edwards 1978, Corollary 3.2) does not apply directly. The result below also requires approximating measures supported on positive orbit pairs, with both marginal densities controlled; those are additional conclusions proved by the sampling construction.

For a finite set \(I\subset\mathbb N\), define a finite Borel measure \(\lambda_I\) on \(X\times X\) as follows. In each coordinate \(i\in I\) use the unnormalized triangular measure \[\mathbf 1_{\{0\le x_i<y_i<1\}}\,dx_i\,dy_i,\] and in each other coordinate use the law of \((x_i,x_i)\) for uniform \(x_i\in[0,1)\). Take the product over all coordinates. Equivalently, one can first normalize the finitely many triangular factors and then multiply the resulting probability measure by \(2^{-|I|}\). In particular, \[\lambda_I(X\times X)=2^{-|I|}.\] Its two marginal densities relative to \(\mu\) are, respectively, \(\prod_{i\in I}(1-x_i)\) and \(\prod_{i\in I}y_i\); they are bounded by \(1\). The measure is carried by \(x\le y\).

We now fix one realization \(\theta\) for which rational independence, Proposition 24, and Proposition 25 all hold, and henceforth write \(\sigma=\sigma_\theta\). The scales and this rotation are fixed before any candidate invariant projection is introduced. For this fixed rotation, each sampling sequence has actual weak convergence against every continuous function; its validity is no longer restricted to the countable tests used to obtain the probability-one event. Thus later choices of continuous tests or compact sets depending on a candidate projection require no further intersection of events in the angle probability space.

The product factor and its weighted shift

We now turn the fixed rotation into an operator in a finite factor. The goal is to express invariance as an identity on ordered pairs, where the sampling result can act. Use the rotation \(\sigma\) fixed at the end of Section 7.2, and write \(\mathcal K=\ell^2(\mathbb Z)\). The operators \(U,\pi(g)\) and their generated algebra \(M\) are those defined at the beginning of the appendix. This is the usual regular representation of a crossed product by a measure-preserving transformation (Anantharaman and Popa, n.d., sec. 1.4 and Proposition 1.4.5); the class of operators \(U\pi(f)\) is also considered in (Dykema and Schultz 2009, sec. 5). For completeness, we verify the properties of \(M\) needed here.

Biased couplings of product coordinates

The measures \(\lambda_I\) give the pair identity when only finitely many coordinates are replaced. We now construct ordered probability laws that give every block opportunities to cross its thresholds. Their endpoint densities remain uniformly bounded in \(L^2(\mu)\), and the crossing estimates remain valid after either entire endpoint is fixed. These are the properties needed for the convolution obstruction.

An ordered coupling with both marginals equal to Haar measure would give \(x=y\) almost surely: in every coordinate, \(y_i-x_i\ge0\) and \(\mathbb E(y_i-x_i)=0\). We therefore allow a controlled change in the endpoint marginals. Fix a block index \(j\ge1\). For \(i\in J_s\), write \(m=m_s\) and set \[ \rho_s(w)=\kappa_s\sqrt{w/m},\qquad w=a_s,b_s. \tag{77}\] Choose the coordinate pair \((x_i,y_i)\) by the following mixture:

• For each of the two labelled widths \(w=a_s,b_s\), include a top component of probability \(\rho_s(w)\). In this component \(x_i\) and \(y_i\) are independent uniforms on \([0,1-w/2)\) and \([1-w/2,1)\), respectively.

• For each labelled width \(w=a_s,b_s\), include a bottom component of probability \(\rho_s(w)\), with independent uniforms on \([0,w/2)\) and \([w/2,1)\), respectively.

• If \(s=j\), include a tagged component of probability \(1/m\), with independent uniforms on \([0,1/m)\) and \([1-1/m,1)\), respectively.

• With the remaining probability \[ q_{j,s}=1-2\rho_s(a_s)-2\rho_s(b_s) -\mathbf 1_{\{s=j\}}/m, \tag{78}\] choose \(x_i=y_i\) uniformly in \([0,1)\).

The width labels remain distinct even if their numerical values agree. The scale bounds imply \[ \frac1m\le a_s,b_s\le\frac14, \qquad \frac{\rho_s(w)}{w}=\frac{\kappa_s}{\sqrt{mw}} \le\kappa_s\le\frac1{100}. \tag{79}\] Since \(m\ge100\), it follows that \(q_{j,s}\ge1-\kappa_s-1/m\ge0.98\).

Make the choices independently over all coordinates. More precisely, let \((\Omega_j,\mathbb P_j)\) be the countable product of the spaces carrying the mixture label and its coordinate pair. Let \(x,y:\Omega_j\to X\) be the endpoint maps, and let \(\nu_j\) be the law of \((x,y)\) on \(X^2\). The labels, including the tags, are thus retained as events on \(\Omega_j\). These are the actual product coordinates of the endpoints. Each component is supported on \(x_i\le y_i\), so \(x\le y\) almost surely.

It remains to establish the pair identity under these probability laws. We first replace a finite prefix of \(x\) by coordinates of \(y\). The resulting law is dominated by a finite sum of the triangular laws \(\lambda_I\), but its diagonal tail retains the biased \(x\)-distribution. Keeping that tail density explicit permits the transfer to infinitely many replacements.

Verifying the convolution obstruction

It remains to exclude invariant projections. We verify the hypotheses of Lemma 17 for the product couplings. The infinite-sum estimate is already contained in that lemma; the task here is to bound the finite coefficients on an event of fixed positive probability and to attenuate every omitted index.

The preceding sections have constructed the factor \(M\), its bounded operator \(T\), and all the couplings, with the scales and angles fixed independently of any projection. Let \(p\in M\) be an orthogonal projection satisfying \((1-p)Tp=0\), and suppose, toward a contradiction, that \(0<p<1\). Use the common invariant Borel conull set for this fixed \(p\) and the cocycles. Equation (76) gives \[0<r(x)=p(x)_{0,0}<1\qquad\text{for }\mu\text{-almost every }x.\] Every coupling used below has absolutely continuous endpoint marginals, so its endpoints belong to this set almost surely. All pointwise operator and cocycle statements below are understood on these sets. Only countably many indices occur, so they may be required simultaneously.

Finite atomic algebras and a diffuse limit

The model also distinguishes the spectral information in finitely many polar conjugates from that in the full orbit. This distinction follows from the explicit weight and the rotation, independently of the convolution argument.

The factor construction, the norm and determinant calculation, the exclusion of invariant projections, and Proposition 31 prove Theorem 23, with \(c_i=C_j\) for \(i\in J_j\). The diffuseness statement is only about \(\mathcal A\); it does not identify \(\mathcal A\) with all of \(\pi(L^\infty(X))\).