Papers

We investigate the relative position of von Neumann subalgebras through the Mashood–Taylor ($\mathrm{d}_{\mathrm{MT}}$) and Kadison–Kastler ($\mathrm{d}_{\mathrm{KK}}$) distances and the interior angle. For each $n\in\mathbb{N}$, we show that the hyperfinite $\mathrm{II}_1$-factor $\mathscr{R}$ contains an uncountable family of pairwise distinct regular $n\times n$ spin model subfactors. In particular, this yields a continuous family $(\mathscr{R}_{\mathsf{H}_\alpha})_{\alpha\in[0,\pi)}$ of $2\times2$ spin model subfactors which, equipped with $\mathrm{d}_{\mathrm{MT}}$, is homeomorphic to the circle group $\mathbb{T}$ and admits a natural topological group structure. We further establish \[ \mathrm{d}_{\mathrm{MT}}(\mathscr{R}_{\mathsf{H}_\alpha},\mathscr{R}_{\mathsf{H}_\beta}) =\vert\sin(\alpha-\beta)\vert. \] We prove that two such spin model subfactors form a commuting square over their intersection if and only if they are maximally distant. More generally, commuting squares of $\mathrm{II}_1$-factors, under natural index conditions, force maximal distance, yielding $\mathrm{d}_{\mathrm{KK}}=1=\mathrm{d}_{\mathrm{MT}}$. We also show that diffuse subalgebras which are orthogonal in the sense of Popa are maximally distant. In contrast, no two members of the above spin model family are Popa-orthogonal. Nevertheless, maximal distance between two such subfactors implies that their interior angle over the intersection is $\pi/2$, highlighting the distinction between angle orthogonality and Popa orthogonality. Finally, in $L(\mathbb{F}_2)=L(\langle a,b\rangle)$, for $u\in L(\langle b\rangle)$, we obtain \[ \mathrm{d}_{\mathrm{MT}}(L(\langle a\rangle),uL(\langle a\rangle)u^*) =\sqrt{1-\vert\tau(u)\vert^4}, \] and construct maximally distant masas not arising from subgroups of $\mathbb{F}_2$.

For a strictly ergodic action of a countable discrete group with Ozawa's property PHP on the Cantor set, we prove that dynamical comparison implies complete selflessness of its reduced crossed product. Therefore these crossed products have stable rank one, strict comparison, a unique quasitracial state, and real rank zero. Applications include probability-measure-preserving Cantor actions of C*-simple acylindrically hyperbolic and C*-simple linear groups for which an infinite amenable subgroup acts strictly ergodically. The results also apply to subodometers and infinite regular Toeplitz subshifts of residually finite groups with PHP.

We construct maximal subgroups of topological full groups and their commutator subgroups using factor maps and wide open subgroupoids of minimal effective ample groupoids with Cantor unit spaces, assuming almost finiteness or pure infiniteness. For two-to-one coverings and prime factor maps with principal target and fibers of cardinality at most two, surjectivity and injectivity of the same induced map on full-group abelianizations characterize maximality of the factor full group and factor commutator subgroup, respectively. For subgroupoids arising from prime-order cyclic actions free on the unit space, the normalizer of the smaller commutator subgroup in the ambient full group is a semidirect product, maximal exactly when the inclusion induces a surjection on full-group abelianizations. In both constructions, the embedded commutator subgroup has a unique maximal overgroup in the ambient commutator subgroup, namely its normalizer. For Cantor minimal systems, the factor construction yields partition stabilizers of the type in the Grigorchuk-Vorobets conjecture. The subgroupoid normalizers act topologically primitively. Groupoid homology makes the criteria computable for these systems and shifts of finite type. Examples include finitely generated nonsimple and locally finite simple maximal subgroups of infinite index in finitely generated infinite simple amenable groups.

Let $X=G/H$ be a real reductive symmetric space. We consider the support of the Plancherel measure for the regular representation of $G$ on $L^2(X)$, and its Fell topology. We first recall how the Plancherel formula for $X$ (known from work of Delorme and van den Ban–Schlicktrull, among others) yields a description of the support in terms of the discrete spectra of smaller symmetric spaces. We then point out that this description simplifies for a class of real symmetric spaces $X$ which we call well-tempered. These are the spaces whose tangent space at the identity coset admits a Cartan subspace whose centralizer in the Lie algebra of $G$ is abelian. Well-tempered spaces are tempered in the sense of Benoist and Kobayashi. Under mild assumptions on the disconnectedness of $G$, we give a description of the support of the Plancherel measure for well-tempered symmetric spaces in terms of characters of Cartan subgroups of $G$, and describe the Fell topology of the support. If $G$ is a complex group and $H$ a real form, then the support decomposes as a union of explicit quotients of relative continuous-parameter spaces by finite Weyl groups. As an application, we describe the $C^*$-algebra associated to the regular representation of $G$ on $L^2(X)$, and its $K$-theory.

It is shown that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb{F}_2)$ is singly generated, where $\mathbb F_2$ is the free group on two generators. With respect to the generator constructed in the paper, the topological free entropy dimension of $\mathrm{C}_r^*(\mathbb{F}_2)$ is shown to be $2$.

We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu–Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid $\mathrm{C}^*$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu–Renault systems are constructed, each modelling the Cuntz algebra $\mathcal{O}_2$. The $\mathrm{C}^*$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into $\mathcal{O}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into $\mathcal O_2\otimes\mathcal K$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a $^*$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit $^*$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.

We show that real rank zero is not preserved under matrix amplifications of inclusions. This yields real rank zero inclusions that do not approximately factor through real rank zero C*-algebras, thereby resolving two questions of Gabe and Neagu.

We investigate the classification of $C^*$-algebras generated by pairs of isometries $(V_1, V_2)$ satisfying $V_1V_2=qV_2V_1$, where $q=e^{2\pi i\theta}$ with $\theta$ irrational, up to unitary equivalence. Using a functional model that realizes pure $q$-commuting pairs on vector-valued Hardy spaces in terms of a uniquely associated triple $(\mathcal{F}, P, U)$ consisting of Hilbert space $\mathcal{F}$, an orthogonal projection $P$ and a unitary operator $U$, we analyze the internal structure of $C^*(V_1,V_2)$ via an extension of the $C^*$-algebra generated by its minimal $q$-commuting unitary extension by the defect ideal $\mathcal{I}_{\mathcal{D}} =\langle [V_1^*, V_1],[V_2^*, V_2]\rangle$. For pairs that are essentially doubly $q$-commuting, we show that $PUP|_{\operatorname{ran} P}$ is semi-Fredholm and prove that $|\operatorname{ind} PUP|_{\operatorname{ran} P}|$ is invariant under unitary equivalence of the corresponding $C^*$-algebras, thereby providing an explicit criterion for distinguishing non-equivalent pairs. This work extends the seminal 1978 results of Berger, Coburn, and Lebow for commuting isometries, while addressing the key technical nuances of this noncommutative setting: most notably, navigating the failure of Gelfand theory due to noncommutativity and replacing the classical Toeplitz extension by an extension of the noncommutative torus by the ideal of compact operators.

We study KMS and ground states of the tame $C^*$-algebras $\mathcal{O}_{m,n}$ of Ara, Exel and Katsura under the dynamics $\sigma_t(s_i)=m^{it}s_i$, $\sigma_t(t_j)=n^{it}t_j$. Normalized restriction to the full corner $A=q\mathcal{O}_{m,n}q$, generated by the partial isometries $v_{ij}=s_i^{*}t_j$ of common energy $\log(n/m)$, is at each inverse temperature $\beta$ an affine weak-$*$ homeomorphism of the two systems. Let $m,n\ge 2$. When $m=n$ the corner dynamics is trivial, so the KMS simplices are identified with trace simplices. When $m\neq n$, KMS states occur only at $\beta=1$, and there they are not unique. Conformal branching models surject the KMS$_1$ simplex onto a transportation polytope of dimension $(m-1)(n-1)$, whose vertices lift to extremal, hence factorial, KMS$_1$ states, the fibre over each vertex being itself the KMS$_1$ simplex of a quotient of $A$.

We introduce a dynamical version $\operatorname{rc} (A, \alpha)$ of the radius of comparison $\operatorname{rc} (A)$ of a unital C*-algebra, based on the dynamical Cuntz semigroup. We also give an intrinsic ordered semigroup definition, which agrees with $\operatorname{rc} (A, \alpha)$ when $A$ is residually stably finite, and is lower semicontinuous for equivariant direct limits with injective unital maps. We construct actions $\alpha$ of $G = \mathbb{Z} / 2 \mathbb{Z}$ on simple unital AH~algebras for which $\operatorname{rc} (A, \alpha)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (A) / \operatorname{card} (G)$, and actions $\alpha$ of a finite group $G$ on unital C*-algebras for which $\operatorname{rc} (A, \alpha)$ lies strictly between $\operatorname{rc} (A)$ and $\operatorname{rc} (C^* (G, A, \alpha))$. For a minimal action of a countable discrete group $G$ on a zero dimensional compact metrizable space $X$, we prove that $\operatorname{rc} (C (X), \alpha) = 0$ if and only if the action has dynamical comparison as defined by Kerr. For finite group actions on simple unital stably finite C*-algebras, assuming the weak tracial Rokhlin property, we get $\operatorname{rc} (A, \alpha) \leq \operatorname{rc} (A) / \operatorname{card} (G)$, and assuming weak tracial strict approximate innerness, we get $\operatorname{rc} (A, \alpha) = \operatorname{rc} (A)$.

This note record a finitely generated extension of Caprace's example of a non-inner-amenable group is also not properly proximal, which answers the original question by Boutonnet, Ioana and Peterson. The extension is an explicit ICC subgroup of $\operatorname{GL}_4(\mathbb{F}_2(t))$, which has no property (T). Failure of proper proximality follows from the work of Ishan-Peterson-Ruth and the passage of proper proximality to co-amenable subgroups. Non-inner-amenability follows from a property-(T) subgroup and an elementary centralizer computation.

A $C^*$-category is to a $C^*$-algebra what a groupoid is to a group. An example is the category of Hilbert spaces and bounded linear maps, and indeed, every $C^*$-category can be represented as a sub-$*$-category of this. Analogously, a $W^*$-category is a $C^*$-category where every object has a predual, and we might expect a representation theory on Hilbert spaces where the resulting morphism spaces are $\sigma$-weakly closed. We identify a gap in the literature here, stemming from the (perhaps surprising) fact that a $C^*$-algebra can have a non-isometric predual and yet not be a $W^*$-algebra. Using the theory of dual TROs (ternary rings of operators) we repair this gap, and along the way, also give further justification to arguments in the literature about dual TROs. Motivated by the theory of Dual Banach Algebras, we offer an alternative axiomatisation of $W^*$-categories where we allow non-isometric preduals of the hom spaces, but require that they are additionally bimodules in a certain sense. This involves the theory of self-dual Hilbert $C^*$-modules.

We construct a function $\Phi $ such that the orbit under the representation of PSL(2,Z) is an orthonormal basis for the Bergman space with weight $\alpha =12$. Moreover, we show that $\Phi$ is effectively computable as a holomorphic function on the upper half-plane (in the precise sense of computable analysis), by providing an effective procedure. This constructs a wandering and tracelike vector for PSL(2,Z), whose abstract existence was proved by Sir Vaughan Jones in his last paper, where the corresponding construction was left as a problem. The function is built using an orthonormalization and modularization method, and it displays modular reminiscencies, despite not being modular itself. provides a computable implementing vector for the abstract anti-isomorphism between the von Neumann algebra $M_{12}(\Gamma )$ and its commutant, which is generated, in Rădulescu's sense, by cusp-form Toeplitz operators, while Voiculescu's results provide a random matrix model for $M_{12}(\Gamma )$.

We prove that $\mathrm{C}^*$-simple groups which admit a faithful extreme boundary are $\mathrm{C}^*$-selfless. More precisely, we establish complete selflessness of the canonical inclusion of the reduced group $\mathrm{C}^*$-algebra into the reduced crossed product. In particular, the Higman–Thompson groups and the Brin–Thompson groups are $\mathrm{C}^*$-selfless.

In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the $p$-logarithmic Sobolev constant vanishes at $p=1/2$ and a correlated graph-based example for which the $p$-logarithmic Sobolev constant tends to $-\infty$ as $p\to 0^+$. These examples obstruct a positive uniform reverse-hypercontractivity rate on $(0,1)$. By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for $1/2< p\leq q < 1$ with explicit sufficient times. For each fixed $\delta>0$, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on $[\delta, 1]$ and hence quantum reverse hypercontractivity for $\delta< p\leq q<1$ with fixed $0<\delta<1/2$. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for $\delta< p\leq q< 1$ with fixed $0<\delta<1$ for connected graphs when $n \geq 3$.

We study the Kadison–Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In \(\mathbb{M}_2(\mathbb{C})\), we obtain an explicit formula for the Kadison–Kastler distance between two masas and show that every value in \([0,1]\) is attained by \(\mathrm{d}_{\textrm{KK}}(\Delta,u\Delta u^*)\). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every $v$ in the groupoid normaliser of $\Delta$, the Kadison–Kastler distance between \(\Delta\) and \(v\Delta v^*\) assumes either its minimal or maximal possible value. We also establish the identity \[ \mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sin\alpha(\mathcal{A},\mathcal{B}) \] for any two intermediate subalgebras \(\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})\). Most notably, for any unitaries \(u,v\in\mathbb{M}_2(\mathbb{C})\), we establish the equivalence of maximal Kadison–Kastler distance, maximal interior angle, maximal Connes–Størmer modified entropy, and minimal Popa probabilistic index: \[ \mathrm{d}_{\textrm{KK}}(u\Delta u^*,v\Delta v^*)=1 \iff h(u\Delta u^*,v\Delta v^*)=\log 2 \iff \alpha(u\Delta u^*,v\Delta v^*)=\frac{\pi}{2} \iff \lambda(u\Delta u^*,v\Delta v^*)=\frac{1}{2}. \] For general \(\mathbb{M}_n(\mathbb{C})\), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form \(u\Delta^{(n)}u^*\) and \(v\Delta^{(n)}v^*\). As a consequence, the interior angle is \(\frac{\pi}{2}\) precisely when the relative unitary \(u^*v\) is a Hadamard unitary. Finally, we show that every value in \(\left[0,\frac{\pi}{2}\right]\) is attained as the interior angle between a pair of masas in \(\mathbb{M}_n(\mathbb{C})\).

For finite groups $G$, we discuss the enrichment of equivariant $KK$- and $E$-theory in genuine $G$-spectra.

We introduce a class of $2d$ quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.

Let $0 \xrightarrow{}I \xrightarrow{ } A \xrightarrow{ }A/I \xrightarrow{} 0$ be a short exact sequence of C*-algebras. In this paper, we prove that if $\dim({\rm Cu}(I))\leq n$ and $\dim({\rm Cu}(A/I))\leq m$, then $\dim({\rm Cu}(A))\leq n+m+1$, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by $I$ and $A/I$ are inherited by the extension algebra $A$. As an application, we provide an alternative proof of equivalence that $A$ is pure if and only if $I$ and $A/I$, a result originally proven by Perera, Thiel, and Vilalta.

We characterise C*-irreducible regular C*-inclusions using a number of different conditions considered by different authors. In particular, we show that all C*-irreducible regular inclusions $A\subseteq B$ are modelled by outer Fell bundles $(B_{g})_{g\in G}$ over discrete groups with a simple unit fibre $A=B_1$. In this case, we prove a bijection between intermediate C*-algebras $A\subseteq C \subseteq B$ and subgroups $H$ of $G$. This extends the Galois correspondence for reduced crossed products by discrete group actions established by Cameron-Smith. We relate it to the Galois correspondences of Izumi and Mukohara for fixed-point algebras of actions of compact abelian groups, and the mixed inclusion of a fixed-point algebra in a reduced crossed product considered by Echterhoff-Rørdam. In addition, using a recent result of Geffen-Ursu, we show that the inclusion of a fixed-point subalgebra $A\subseteq B$ of an action of $\mathbb{T}$ or $\mathbb{Z}/p$ for a square-free number $p>0$ is aperiodic if and only if $A$ detects ideals in $B$. We apply this to give examples of C*-irreducible inclusions coming from Cuntz-Pimsner algebras, including crossed products by endomorphisms or transfer operators. In particular, we characterise when a core subalgebra of a graph C*-algebra is C*-irreducible. Lastly, we show that a general regular topologically graded C*-inclusion $A\subseteq B$ is aperiodic and has a unique pseudo-expectation provided $A$ detects ideals in all intermediate C*-algebras of $B$. This partially answers a question by Pitts-Zarikian.

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. A subalgebra $\mathfrak{A}$ in $\mathcal{B}(\mathcal{H})$ has the left (resp.\ right) partial factorization property if for any invertible operator $S\in\mathcal{B}(\mathcal{H})$, there exists an isometry (resp.\ a co-isometry) $U\in\mathcal{B}(\mathcal{H})$ such that $U^*S, S^{-1}U\in\mathfrak{A}$. We show that if $\mathfrak{A}$ is weak operator topology closed with the left (resp.\ right) partial factorization property, then $\mathfrak{A}$ is the nest algebra associated with its invariant subspace lattice. In particular, if $\mathfrak{A}$ is transitive, then $\mathfrak{A}=\mathcal{B}(\mathcal{H})$. This gives a positive answer to Question 6.3 raised by B.V.R. Bhat and M. Kumar in Publ. Res. Inst. Math. Sci. 60(2024), 507–537.

We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt3$. If $\Phi_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $\Phi_*(q)=2/(\sqrt3-q)+O(1)$ as $q\uparrow\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér–Riesz factorisation.

We give a proof that the full group $C^*$-algebra of a countable discrete amenable group is strong NF. The representation-theoretic construction uses the exact sequence whose kernel is the hyper-FC radical and whose quotient is the maximal ICC quotient. Extreme invariant traces on the radical induce factorial representations of the group. A Choquet decomposition of the regular radical trace selects a separating family whose induced traces are supported on the ordinary FC-centre. Exact twisted crossed-product quotients and a sequential UCT argument make their images quasidiagonal. Faithful irreducible replacements then supply the projections required for inner quasidiagonality. We also give a direct trace-decomposition variant and identify the steps shared by the two proofs.

We construct a separable unital residually finite-dimensional $C^*$-algebra $A$ whose maximal tensor square contains a proper isometry. In particular, $A$ is quasidiagonal whereas $A\tmax A$ is not. The algebra is generated by finite regular representations obtained from congruence quotients of an explicit countable affine group. A property~$(T)$ subgroup determines a diagonal Kazhdan projection in the maximal tensor square. Conjugation by a dilation makes this projection equivalent to a proper subprojection. Strictness is detected by a quasi-regular representation whose two factor restrictions descend to the specified algebra $A$. We give the norm estimate needed for this descent and construct the proper isometry explicitly.

We construct in ZFC a unital, simple, infinite-dimensional $C^*$-algebra whose nonzero irreducible representations form a single unitary equivalence class, but which admits a faithful representation on a separable Hilbert space. The algebra contains a unital copy of the canonical anticommutation relation (CAR) algebra, and the normalized CAR trace has a unique extension among all states. This extension is tracial, and its GNS representation is separable and faithful, with weak closure the hyperfinite $\mathrm{II}_1$ factor. In contrast, every nonzero irreducible representation acts on a Hilbert space of density $2^{\aleph_0}$. The construction separates the added unitaries into shell terms, controlled by finite CAR relations, and rank-one defect terms. These relations determine every irreducible representation of the generated algebra and every extension of the CAR trace. The algebra has norm density $2^{\aleph_0}$; consequently, the continuum hypothesis (CH) is equivalent over ZFC to the existence of a counterexample to Naimark's problem of norm density $\aleph_1$.

We exhibit explicit ICC subgroups of $\SL_7(\mathbb Q)$ with Kazhdan's property $(T)$ that is not properly proximal.

In this paper, we prove that the coarse Novikov conjecture holds for finite products of bounded-geometry proper metric spaces whose factors admit fibred coarse embeddings into possibly different target spaces. A key ingredient is the construction of suitable coarsely proper algebras adapted to product spaces. Our other main tool is an iterated approach to relative higher index theory. By constructing the corresponding multi-stage Roe algebras and index maps, we establish a reduction theorem that bridges these iterated relative statements with the global coarse Novikov conjecture.

Liu and Ruan showed that a separable operator space has the completely bounded approximation property if and only if it admits a completely bounded frame. We show that, even for Hilbertian operator spaces, such a frame need not give rise to a completely bounded basis. More precisely, we construct a separable Hilbertian operator space with the $1$-CBAP but with no completely bounded basis. Our construction relies on a theorem of Oikhberg that provides a structural description of the completely bounded maps on a suitable operator space.

We propose a notion of relative coarse rigidity for $\sigma$-finite von Neumann algebras through modular derivations and their associated GNS-symmetric quantum Markov semigroups, extending Peterson's framework of $L^2$-rigidity to the nontracial setting. Our main result shows that the absence of relatively amenable summands forces the associated deformations to converge on relative commutants in ultrapowers. For diffuse von Neumann algebras with separable predual, this yields a dichotomy between the presence of an amenable summand and rigidity in terms of central sequences. These results give a novel unified approach for proving indecomposability properties such as fullness, relative $\omega$-solidity and relative stable solidity. Our methods apply to examples such as amalgamated free products, operator-valued free Araki-Woods factors and cocycle crossed products.

Dixmier asked in 1967 whether the spectra of simple separable non-type-I $C^*$-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel $\sigma$-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for $\omega_2$ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal. The Borel-definable formulation, which requires isomorphisms to admit Borel lifts between irreducible representations, has a negative answer in ZFC. For the CAR algebra $A$ and a simple separable exact crossed product $B$ associated to $\mathrm{SL}_3(\mathbb{Z})$, there is no Borel-definable injection $\widehat B\to\widehat A$. Nevertheless, all separable nuclear non-type-I spectra are Borel-definably isomorphic. We also determine the answer under several choiceless hypotheses. The axiom of determinacy together with dependent choice gives a stronger negative conclusion: the spectrum of $B$ does not even inject as a set into the spectrum of $A$. The symmetric Solovay model gives the same conclusion. Over ZF, a wellordering of the reals of order type $\omega_1$ suffices for a positive answer. The finite-permutation Feferman–Levy model gives another positive answer for the full sigma-generated Borel structures. These comparisons concern the quotient structures themselves, without measure completion.