Papers

We introduce the crystal groupoid of a compact semisimple Lie group $K$; it is the Deaconu-Renault groupoid of a higher-rank analogue of a subshift of finite type on an inverse limit of Kashiwara crystals. We show that its Steinberg algebra and its $C^*$-algebra realise the crystal limits of the polynomial function algebra of $K$ and its $C^*$-envelope, respectively, in the sense of Matassa and the second author. We show that the orbit space of the crystal groupoid is in bijection with the Weyl group of $K$, with topology determined by the Bruhat order on the Weyl group. We also define a crystal groupoid for flag manifolds and related Poisson homogeneous spaces, and prove the analogous structural results.

We prove the Gromov-Hausdorff convergence of the state spaces, equipped with Connes' distance function, of the operator systems associated with spectral truncations for noncommutative $n$-tori, $n\ge 2$. The action of the $n$-torus on the noncommutative $n$-torus, together with an appropriate use of Bochner integrals, enables us to construct an $C^1$-approximate order isomorphism that establishes the convergence. The earlier convergence result for the classical case of tori [20] follows as a special case of our result. We conclude by studying the $C^*$-envelopes and propagation numbers of the truncated operator systems and show that the structural properties of the truncated operator systems for tori persist under deformation.

We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii \rho_{E_1} and \rho_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.

We establish duality relations for conditional entropies relative to subalgebras of arbitrary von Neumann algebras, including the Umegaki relative entropy as well as the sandwiched- and Petz-Rényi divergences. Even in finite dimensions, these results extend the familiar dualities for conditional entropies from tensor-product subsystems to general subalgebras, while also lifting the framework to the fully infinite-dimensional setting. As an application, we establish an infinite-dimensional extension of quantum Stein's lemma when the alternative hypothesis is the state space of a finite-index subalgebra, which gives an operational interpretation of the relative entropy to a subalgebra as the optimal asymptotic error exponent in composite quantum hypothesis testing-the first beyond finite dimensions.

Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct $q+1$ bases in dimension $d = q(q+1)$ for every prime power $q\equiv3\pmod4$. This count grows as $\sqrt d$ and cannot be exceeded by tensor products of smaller sets.

We call a countable discrete group cocycle-hyperlinear if all its twisted group von Neumann algebras are Connes-embeddable. We show that the class of cocycle-hyperlinear groups includes extensions of treeable groups by amenable groups and central extensions of SL(n,Z), GL(n,Z) and PSL(n,Z) for n greater than 1. We also show that this class is closed under taking direct products, free products, amalgamated free products with amenable amalgams, HNN extensions by amenable groups, as well as certain quotients, extensions, and overgroups.

Open quantum systems can rapidly lose most microscopic information, leaving only a few degrees of freedom to govern their long-time dynamics. Classical Markov state models (MSMs) describe such metastable dynamics as transitions among a few representative phases and are widely used to reduce complex-system dynamics in condensed matter and chemical physics. In quantum systems, however, phase labels alone are insufficient when coherence persists between metastable states. Even when the slow modes are known, their spectral projection need not produce valid quantum states. We construct quantum Markov state models (QMSMs) that describe the surviving information and its evolution on a small physical state space containing classical sectors and quantum matrix blocks. We quantify metastability by assuming that the evolution channel $\mathcal C$ changes little when applied a second time, with sufficiently small defect $\eta=\|\mathcal C^2-\mathcal C\|_\diamond$, and that the number of slow degrees of freedom is bounded independently of the full system size. Under these assumptions, we construct compression and reconstruction channels whose composition recovers every reduced state exactly, while the reverse composition gives an exactly idempotent channel approximating $\mathcal C$, answering Kitaev's exact-rounding question [Kit25]. Our constructions give an optimal diamond-norm bound of $\mathcal{O}(\eta^{1/3})$ on all input states, as well as an improved bound of $\mathcal{O}(\eta^{1/2})$ on metastable states prepared by $\mathcal C$, with constants depending only on the slow dimension. The reduced transition channel can be iterated to predict the microscopic dynamics with controlled error. We illustrate the QMSM through a weakly driven dissipative spin chain supporting either a metastable logical qubit or long-lived classical phases.

In this note, we study complete and quantum Gromov-Hausdorff convergence of operator systems obtained using smoothings of C*-algebras, where our smoothings arise from the action of a compact quantum group on the C*-algebra. Specifically, we show how if $\mathbb{G}$ is a co-amenable compact quantum group acting on unital C*-algebra $\mathcal{A}$, and both are equipped with suitable quantum metrics, bounded approximate identities in $L^{1}(\mathbb{G})$ may be used for approximations of the original space $\mathcal{A}$ under the complete and quantum Gromov-Hausdorff distance. We finish by presenting some applications to the convergence of spectral truncations of compact groups and connections to previous results.

We compute $K$-theoretic invariants for four three-dimensional operator systems to illustrate the general theory. We find that in some cases the invariants distinguish the operator system from its $C^*$-envelope, while in others they do not. We also explore extensions of the theory.

We compute $K$-theoretic invariants for several finite-dimensional operator systems. These include the Toeplitz operator system and its dual (the Fejér-Riesz operator system), as well as operator systems associated to tolerance relations, aka graph operator systems. A variety of techniques is applied, adapted to each of these cases, ranging from a generalization of Carathéodory's factorization result for block Toeplitz matrices, to Wiener-Hopf factorization and Schur complements. In most instances we find that the invariants coincide with their analogues for the $C^*$-envelope, except for the $K_1$-invariants of the Fejér-Riesz operator system.

Previous work of Vaughan Jones built a bridge between cusp forms and random matrices using the language of von Neumann algebras. Specifically, for a discrete subgroup of $\operatorname{PSL}_2(\mathbb Z)$, there is a natural von Neumann algebra associated to it. Jones showed that for any "tracelike” vector, there is a corresponding (anti)-isomorphism between this algebra and its commutant. Jones also showed that such vectors exist. As he noted, this connection is not useful without an explicit formula, which he posed as an open question. Here, we resolve Jones' question and discuss its context for a broad number theory audience. As we were finalizing this preprint for submission, we learned of an independent paper of Abreu. Both papers employ a closely-related polar normalisation of a kernel orbit to obtain the Jones tracelike vector at the critical parameter. The novelty of the present work lies in its extension to all $1<s\leq13$ and in the detailed analysis of the resulting von Neumann algebra and cusp form correspondences. This paper also arose out of a near solution of the authors from several years ago, which discussions with ChatGPT helped us finalize into a full solution. The relation to Abreu's work, and the exact nature of our use of AI, are detailed in the introduction.

At what rate does the von Neumann entropy of an ensemble of quantum states change under Hamiltonian evolution of its constituents? Bravyi proposed the small incremental mixing conjecture controlling the mixing rate of a binary ensemble $\{(1\!-\!p,\rho_1),(p,\rho_2)\}$ by $c\,\|{H}\|h_{2}(p)$ (with the binary entropy $h_{2}$). The proof of Bravyi's conjecture was subsequently reduced to the matrix inequality $\|[A,\log B]\|_{1}\!\leq\! c\,h_{2}(p)$ for positive trace-class operators $A\!\leq\! B$ with $\text{Tr} A\!=\!p$ and $\text{Tr} B\!=\!1$ on separable Hilbert spaces by Mariën, Audenaert, Van Acoleyen, and Verstraete, and they conjectured the optimal constant to be $c\!=\!1$. Here, I prove the latter conjecture using an exact integral representation of the commutator $[A,\log B]$ obtained from the operator layer cake theorem, due to Cheng and Liu. The result gives a sharp dimension-independent limit on how rapidly unitary evolution of one component can change the entropy of a binary quantum ensemble. It follows that mixing rates satisfy small incremental mixing with the optimal constant, and entangling rates of bipartite Hamiltonians are bounded by $(2\log d\!+\!1)\|H\|$. Moreover, the result corrects a conjecture by Lieb and Vershynina for mixing rates of general ensembles.

Both the uniform Roe algebras and the quasi-local algebras encode the large scale geometry of metric spaces. Whether these algebras coincide is a question which goes back to Roe and has only been recently solved by Ozawa. We propose a dynamical point of view on this problem. Given a set $X$, every map $h\colon X\to\mathbb{R}$ induces a one-parameter group $\sigma_h$ of automorphisms of $\mathcal{B}(\ell_2(X))$, given by conjugation by the diagonal unitaries $e^{ith}$, and we show that the operators which are continuity points of $\sigma_h$ are precisely the uniform Roe algebra of the pseudo-metric induced by $h$. If we moreover assume that $X$ is a uniformly locally finite metric space and $h$ is allowed to range over all coarse maps, this gives dynamical characterizations of both algebras: $\mathrm{C}^*_{ql}(X)$ consists exactly of the operators which are continuous for all such flows, while $\mathrm{C}^*_u(X)$ consists exactly of norm limits of those which are analytic. The regularity conditions lying between continuity and analyticity then give rise to $\mathrm{C}^*$-algebras between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$ which are invariant under bijective coarse equivalence, and we show that this scale is not degenerate: if $X$ is a coarse disjoint union of expander graphs, the algebra generated by operators which are analytic on a strip for every diagonal flow sits strictly between $\mathrm{C}^*_u(X)$ and $\mathrm{C}^*_{ql}(X)$.

Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every pair of elements, one from each subalgebra. Writing such a pair as ${\sf D}_n$ and $U{\sf D}_nU^*$ with $U$ unitary, we characterize finite free position, for either additive or multiplicative convolution, by the condition $|\det U[I,J]|^2=\binom{n}{r}^{-1}$ for every $1\le r\le n$ and all $I,J$ with $|I|=|J|=r$. We show that this condition holds if and only if $n\le3$ and $\sqrt{n} U$ is a complex Hadamard matrix. To quantify the failure of exact realization for $n\ge 4$, we introduce the uniform-minor discrepancy $\delta_r(U)$. We identify it with the mean-square error in the $r$-th coefficient of finite free multiplicative convolution for two diagonal matrices whose diagonal entries are independent and uniformly distributed on the unit circle. We establish the symmetry $\delta_r(U)=\delta_{n-r}(U)$ and the monotonicity $\delta_1(U)\le\delta_2(U)\le\cdots\le \delta_{\lfloor n/2\rfloor}(U)$. For flat unitaries, we derive an explicit formula for $\delta_2$, yielding $\delta_r(U)\ge\frac{n-3}{2n}$ for $2\le r\le n-2$. Equality for $r=2$ holds precisely when the entrywise square of $\sqrt{n}U$ is also complex Hadamard.

Given any tracial von Neumann algebra $\mathcal{A}$, the first author has defined a unitary tensor functor $\Phi:{\rm Rep}(\mathcal{Q}_{\rm aut}(\mathcal{A}))\to {\rm Bimod}(\mathcal{A})$ in his paper [Gos24]. In this paper, we consider $\mathcal{A}$ to be the underlying von Neumann algebra of $C^*(S_3)$ and show that given any DQG $\mathcal{Q}$, which contains $C^*(S_3)$ as a quantum subgroup and has an outer action $\tilde{\Phi}$ (in the sense of Definition 3.4}) on ${\rm Bimod}(\mathcal{A})$, which agrees with $\Phi$ when restricted on ${\rm Rep}(C^*(S_3))$, is monoidally equivalent to the DQG $C^*(S_3)$. Furthermore, given any DQG $\mathcal{Q}$ with an outer action $\Gamma:{\rm Rep}(\mathcal{Q})\to {\rm Bimod}(\mathcal{A})$, such that $ {\rm Out}(\mathcal{A})\subseteq {\rm Im}(\Gamma)$, is either monoidally equivalent to Out$(\mathcal{A})$ or monoidally equivalent to $C^*(S_3)$ itself.

Let $A$ and $B$ be strongly Morita equivalent $C^*$-algebras, implemented by a nonzero imprimitivity bimodule $X={}_AX_B$ with its canonical operator-space structure. Denote by $Z_X$ their common multiplier centre, canonically identified through $X$. The coefficient action gives a complete contraction $\Theta_X:A\otimes_{Z_X,h}B\to\operatorname{CB}(X)$, where $A\otimes_{Z_X,h}B$ is the central Haagerup tensor product obtained by balancing over $Z_X$ and $\operatorname{CB}(X)$ is the space of completely bounded maps on $X$. We show that $\Theta_X$ is injective exactly when every Glimm ideal of $A$ is $2$-primal, and isometric, equivalently completely isometric, exactly when every such ideal is primal. For each positive integer $\ell$, norm preservation for sums of at most $\ell$ elementary tensors is equivalent to $(\ell^2+1)$-primality of every Glimm ideal. These conditions may equivalently be imposed on $B$. This extends the corresponding theorems of Somerset and Archbold–Somerset–Timoney to imprimitivity bimodules. We also show that, after matrix stabilization, the optimal constants bounding the central Haagerup norm by the completely bounded norm coincide for $X$, $A$ and $B$, at every finite tensor length and on the completed products, and agree with the corresponding constants after compact stabilization.

We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.

Let $B_\infty$ and $P_\infty$ be the braid group and the pure braid group on infinitely many strands. We show that the inclusion $C^*_r(P_\infty)\subseteq C^*_r(B_\infty)$ is $\mathcal{Z}$-stable and selfless. Hence every intermediate inclusion and every intermediate $C^*$-algebra is $\mathcal{Z}$-stable and selfless. The intermediate $C^*$-algebras are natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^*$-algebras, and they are singly generated. The proof of $\mathcal{Z}$-stability uses a simple relative centralizer criterion. We also show that $C^*_r(\mathbb{F}_\infty)\subseteq C^*_r(B_\infty)$ is a selfless inclusion, where $\mathbb{F}_\infty$ is the free group of braids in which only the first string moves. All results hold for reduced twisted group $C^*$-algebras. Selflessness of $C^*_r(B_\infty)$ itself already follows from work of Ozawa. On the other hand, $\mathcal{Z}$-stability fails for all twisted group $C^*$-algebras of groups that are not inner amenable. A variant shows that it fails for all twisted group $C^*$-algebras of $B_n$ with $n$ finite, and for the twisted crossed products from Artin's representation of $B_n$. Finally the method applies to other increasing unions of groups, such as $\mathrm{SL}_\infty(\mathbb{Z})$, whose reduced group $C^*$-algebra is $\mathcal{Z}$-stable.

We show that every UCP extension problem for a finite dimensional operator system can be reduced to three dimensions after finite matrix amplification. The set of extensions is preserved by an affine bijection. The matrix size is of order $\sqrt{d}$, where $d$ is the dimension of the original system. This also gives a three dimensional (smallest) counterexample to Arveson's hyperrigidity conjecture.

We show that the optimal commuting operator correlation for a variant of Coladangelo's two-player generalized nonlocal game, based on the embezzlement of entanglement, is a commuting operator self-test. Our result gives the first correlation self-test for an exact entanglement embezzlement protocol in the commuting operator framework. Because the unique optimal correlation necessitates the exact embezzlement of entanglement, the correlation is unrealizable using quantum tensor-product models, even if the local spaces are allowed to be infinite dimensional. As such, our result provides an example of a commuting operator self-test where the Hilbert space shared by the two parties cannot be factored into a tensor-product of local spaces that is compatible with the local measurement operators. Our main proof technique relies on Harris' unique abstract state characterization of exact embezzlement of entanglement involving Cuntz algebras.

Hilbert C*-modules whose coefficient C*-algebra is equipped with a finite faithful trace generate Banach Gelfand triples. This provides a natural setting for a function space interpretation of Hilbert C*-modules. In particular, when the modules are finitely generated and projective, their external tensor products interact well with projective tensor products, allowing us to derive abstract kernel theorems for Hilbert C*-modules. As an application, we apply our results to Heisenberg modules and obtain new kernel theorems for time-frequency analysis.

We prove that the relative commutant of a finite unitary tuple in a tracial matrix ultraproduct is internal whenever its conjugation operator has a spectral gap off the commutant. This implies the existence of a non-hyperlinear group.

This is the first in a series of papers extending Roe algebras and coarse index theory to the groupoid-equivariant setting. We introduce some techniques to develop a framework of Roe algebras and their K-theory for spaces equipped with proper actions of étale groupoids. For a fixed étale groupoid G and G-C* -algebra A, we construct a functor KC(-; G, A) from the category of locally compact Hausdorff proper G-spaces and equivariant proper continuous maps to the category of graded abelian groups, which provides a natural receptacle for an equivariant coarse index map. We study the existence of universal modules, an analog of ample modules in non-equivariant setting, and develop a decomposition technique to establish their existence. As an application, we prove that groupoid simplicial complexes satisfying suitable hypotheses admit universal modules.

We argue that the exponential quantum advantage in Shor's algorithm is broken under one-layer depolarizing noise of arbitrary small error rate, by analyzing a resonance breaking phenomenon in noisy quantum circuits. First, we express the distribution of measurements on $n$-qubit strings as the superposition of the $4^n$ wave functions in the Pauli path integral. In the noiseless case, the bit-strings achieving resonant peaks guaranteed a constant rate of successful measurements to factor a large number. Secondly, when the middle layer of the quantum circuit has independent depolarizing noise of rate $\lambda$, for any Hamming weight $d$, we obtain corresponding measurement rate bounded by $2(1-\lambda)^d$ for higher frequency terms and by $O(n^d)/2^{n/2}$ for low frequency terms. The rate approaches to zero as $n$ and $d$ approaches to infinity. Resonance breaking destroys the exponential quantum advantage. Furthermore, we design a classical factoring algorithm in polynomial time $O(n^d)$ to substitute the low frequency contribution.

We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.

We prove that free additive convolution preserves the $k$-convex order for every $k\ge2$, assuming finite $(k-1)$st absolute moments for the compared measures and a finite $\max\{2,k-1\}$th absolute moment for the convolving measure. In particular, the case $k=4$ gives an affirmative answer to a question of Heinävaara concerning fourth-order convolution comparison.

Let $\Gamma$ be a countable discrete group and let $A$ be a unital $\Gamma$-simple $\Gamma$-$C^*$-algebra. We prove that $A\rtimes_r\Gamma$ has the generalized Powers averaging property if and only if the canonical map $A\rtimes_r\Gamma\to I_\Gamma(A)$ is the unique $\Gamma$-equivariant pseudo-expectation. We also show that this averaging property lifts to $I_\Gamma(A)\rtimes_r\Gamma$, with the averaging coefficients still belonging to $A$.

Let $G$ be a reductive group over a number field $F$. We define $C^*_{\rm aut}(G(\mathbb A))$, the automorphic $C^*$-algebra of $G$, to be the $C^*$-algebraic image of the full automorphic representation of $G(\mathbb A)$ on $L^2(G(F)\backslash G(\mathbb A))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $*$-homomorphism \[ C^*_{\rm aut}(G(\mathbb A)) \longrightarrow \bigoplus_{[M,\sigma]} K_{C_0(\widehat{A_M})} \left( \operatorname{Ind}_P^G C_0(\widehat{A_M},H_{M,\sigma}) \right)^{W(G,M,\sigma)}, \] where $[M,\sigma]$ ranges over the associate classes of discrete Levi data. When $G$ is $\operatorname{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^*$-algebras.

We prove that the weak Haagerup constant is multiplicative with respect to direct products, namely $\Lambda_{\mathrm{WH}}(G \times H)=\Lambda_{\mathrm{WH}}(G) \Lambda_{\mathrm{WH}}(H)$ for any locally compact groups $G$ and $H$. Knudby previously proved the submultiplicative inequality and asked whether equality always holds.

We establish maximal inequalities for geometric averages of operator-valued functions in noncommutative \(L^p\)-spaces associated with semifinite von Neumann algebras. For averages over a fixed smooth submanifold of finite type at the parameter origin, we prove local maximal bounds for every \(1<p\leq\infty\). For polynomial parametrizations, we obtain bounds over all positive scales without a finite-type assumption, with constants only depending on the degree and dimensions. We also prove local maximal inequalities for variable hypersurfaces in \(\mathbb R^n\), \(n\geq3\), satisfying a uniform rotational curvature condition, in the range \(p>n/(n-1)\). The finite-type and polynomial estimates rely on a weak type \((1,1)\) inequality for a regularized auxiliary family adapted to non-isotropic dilations. Its proof uses a noncommutative Calderón-Zygmund decomposition based on Cuculescu projections. Interpolation with Fourier-based \(L^2\) bounds recovers the maximal inequalities for the original averages. The variable-hypersurface result uses a separate argument based on oscillatory \(L^2\) estimates and localization. As applications, we obtain some noncommutative maximal ergodic inequalities for trace-preserving actions of \(\mathbb R^n\) (corresponding to the geometric averages considered above) and bilateral almost uniform convergence for normalized ergodic averages.