Papers

Let $\hbar$ be a non-zero real number and put $q= e^{-\pi i \hbar}$. Let $G$ be a semisimple split real Lie group of classical type, with Lie algebra $\mathfrak{g}$ and associated Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$. We show that there exists a locally compact quantum group $(L_{\hbar}(B),\Delta)$ which integrates the Borel part $U_q(\mathfrak{b})$ of the Drinfeld–Jimbo quantized enveloping algebra $U_q(\mathfrak{g})$, viewed as a Hopf $*$-algebra. We further compute the associated Haar weights and modular data for $(L_{\hbar}(B),\Delta)$. Up to an excess Cartan part, the Drinfeld double of $(L_{\hbar}(B),\Delta)$ can be seen as an analytic quantization of the function algebra of the totally positive semigroup $G^+\subseteq G$.

We study a strong pure infiniteness property for extensions, and give a classification result for extensions which contains many (and possibly all) previous cases, under standard and mild assumptions. We also prove a Voiculescu-type theorem decomposition theorem for certain C*-subalgebras of a corona algebra, for which the inclusion map has this strong pure infiniteness property.

This paper studies continuous covariant kernel families over hulls of Delone sets of finite local complexity using graded Wiener-algebra methods. The norm given by the least common weighted summable off-diagonal profile is identified isometrically with a graded $\ell^1$-norm. The resulting Banach $*$-algebras are symmetric, spectrally invariant in the reduced groupoid C*-algebra, and stable under holomorphic functional calculus. Exact physical weights give preservation of polynomial and stretched-exponential classes, including a fixed stretched-exponential rate. A configuration with dense orbit has a faithful regular representation. Invertibility on that configuration therefore gives an inverse family with one profile valid throughout the hull; for repetitive sets every configuration has this property. For polynomial weights of order $s>0$, return-vector counting yields a differential estimate with parameter $\kappa/(\kappa+s)$, where $\kappa$ is a growth exponent of the return vectors and may be taken equal to $d$ for Meyer sets. Combined with established norm-controlled inversion criteria, this gives explicit bounds in the common-profile norm. Explicit bounds are also obtained for stretched-exponential physical weights and for gapped Fermi projections.

Let $G$ be a locally compact group, let $\Gamma$ be a lattice in $G$, and let $\eta$ be a spread-out probability measure on $G$. A Furstenberg discretization of $\eta$ to $\Gamma$ is a probability measure $\mu$ on $\Gamma$ such that every $(G,\eta)$-stationary space is also $(\Gamma,\mu)$-stationary. For cocompact lattices $\Gamma$ and under mild regularity conditions on $\eta$, we construct Furstenberg discretizations satisfying finite moment conditions of arbitrary order with respect to arbitrary prescribed subadditive functions. Our approach relies on a rather general study of differential entropy and Lyapunov exponents in the setting of locally compact groups. We relate Lyapunov exponents to spectral radii in weighted $L^{1}$-algebras and prove vanishing results for weights coming from length functions. Additionally, we prove that if $G$ has the property of rapid decay or the Kunze–Stein property, this yields entropy formulas for stationary actions whose Koopman representation is weakly contained in the left-regular representation. As consequences, we obtain preservation results for Poisson boundaries under discretization and characterize Zimmer amenability of stationary actions in terms of weak containment and maximal Furstenberg entropy.

Let $A$ be a unital, simple, separable, and pure $C^*$-algebra. We prove that, for every $a\in A_+$, the set of positive elements in $A$ having the same Cuntz class as $a$ is path-connected. This may be viewed as a generalization to positive elements of the classical homotopy theory of projection classes and extends a line of results on the topology of constant Cuntz classes that previously required stronger regularity hypotheses to the class of pure $C^*$-algebras, which notably contains all selfless algebras.

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

Consider a minimal and free topological dynamical system $(X, \Gamma)$, where $\Gamma$ is a discrete amenable group. Assume $(X, \Gamma)$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X) \rtimes \Gamma) = \frac{1}{2} \mathrm{mdim}(X, \Gamma).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb Z$-actions and the topological ideas used in that paper.

Let $M$ be a nonzero type III von Neumann algebra with separable predual, and let $N$ be a semifinite von Neumann algebra. We prove that $M_*$ is not isomorphic to a subspace of $N_*$. For $1<p<\infty$, $p\ne2$, we prove that $L_p(M)$ is not isomorphic to a complemented subspace of $L_p(N)$. In particular, the Banach isomorphism class of $L_p(M)$ distinguishes type III algebras from semifinite algebras for $1\le p<\infty$, $p\ne2$.

Let $(\mathcal N,\nu)$ be a finite tracial von Neumann algebra and set $\mathcal{K}_r(P,Q)=\nu((P^{1/2}QP^{1/2})^r)$. We prove that $$ [\mathcal{K}_r(P_i,P_j)]_{i,j=1}^3\ge0, \qquad 1/2\le r\le1. $$ The range $1/2<r<1$ is new even for matrices, and order three is sharp for $1/2\le r<1$. This single mechanism resolves two problems. First, if $(\mathcal R,\omega)$ is finite tracial and $H=[H_{ij}]_{i,j=1}^3\in M_3(\mathcal R)_+$, then $$ [\omega(|H_{ij}|^p)]_{i,j=1}^3\ge0, \qquad 1\le p\le2, $$ settling the Lin–van den Driessche conjecture on Schatten norm compression. Second, for a semifinite tracial von Neumann algebra $(\mathcal M,\tau)$, define $$ d_t(A,B)^2=\frac{\tau(A)+\tau(B)}2- \|B^{t/4}A^{t/4}\|_{2/t}^{2/t}. $$ We prove that $d_t$ is a complete metric on $L^1(\mathcal M,\tau)_+$ for $1\le t\le2$, including the previously open range $1<t<2$ between the Hellinger and Bures endpoints, and that $$ d_t\text{ is a metric on }L^1(\mathcal M,\tau)_+ \quad\Longleftrightarrow\quad t\in[1,2]\ \text{or}\ \mathcal M\text{ is abelian}. $$ Its topology and Cauchy sequences agree with those of $L^1$. The same classification holds for unital tracial $C^*$-algebras and, with the known $t=0$ obstruction, resolves Problem~2 of Komálovics and Molnár.

We establish a geometric rigidity theorem for a class of monomial varieties in finite-dimensional unit balls. Every ball automorphism mapping one such variety onto another is unitary, and whenever such an equivalence exists, it can also be realized by a permutation of the ambient coordinates. As a consequence, we obtain a complete isometric classification of the multiplier algebras of integrally generated complete Nevanlinna–Pick Dirichlet series kernels in terms of their defining weight and frequency data. We also obtain an explicit isometric classification for a class of complete Nevanlinna–Pick kernels on bounded Reinhardt domains.

In this paper, we introduce the crystallization $C(SO_0(2n+1)/SO_0(2n-1))$ of the $C^*$-algebra $C(SO_q(2n+1)/SO_q(2n-1))$. We show that every irreducible representation of $C(SO_0(2n+1)/SO_0(2n-1))$ arises as the limit, as $q \to 0^+$, of a family of irreducible representations of $C(SO_q(2n+1)/SO_q(2n-1))$. As a consequence, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is the universal $C^*$-algebra generated by a finite set of generators and relations. We then construct a chain of unital essential short exact sequences of $C^*$-algebras, enabling us to compute the $K$-groups of $C(SO_0(2n+1)/SO_0(2n-1))$ and determine explicit generators for them. Using the theory of homogeneous $C^*$-extensions, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is isomorphic to $C(SO_q(2n+1)/SO_q(2n-1))$ for every $q \in (0,1)$.

We construct an explicit binary linear system game that separates $C_{qa}$, the closure of the set of finite dimensional quantum correlations, from $C_{qc}$, the set of commuting operator correlations. The game admits a perfect commuting operator strategy, while every correlation in $C_{qa}$ has a success probability strictly less than 1. This provides a concrete counterexample to Tsirelson's problem in its approximation form. The defining system has 1417152 equations in 1889684 variables, with exactly three nonzero coefficients per equation and a single nonzero entry on the right hand side. We also compute the classical value exactly. The complete system is specified in Lean 4, and the separation and exact classical value are formalized using Mathlib.

We prove invariance of the geometric Arveson-Douglas conjecture under Veronese embeddings of projective varieties. This provides new examples of singular varieties satisfying the conjecture. Furthermore, it reduces the conjecture to projective varieties whose defining ideals admit quadratic Gröbner bases, by keeping the same Schatten regularity thresholds. Our results lead to canonical examples of subproduct-systems whose Toeplitz algebras are not $KK$-equivalent to the complex numbers. Finally, we show that Douglas' fundamental class conjecture, asserting that the associated Toeplitz extension implements $\mathbb{T}$-equivariant duality in $KK$-theory, is invariant under Veronese embeddings as well. While the latter result bootstraps the fundamental class conjecture, it also leads to infinitely many counterexamples.

In this paper, we prove that every separable unital AT algebra of real rank zero has complexity rank at most one. In particular, every irrational rotation algebra has complexity rank one. This answers a question of Jaime and Willett.

We study iterated function systems with large overlaps via the Kajiwara-Watatani C*-algebra and its KMS states. Our main contributions are threefold. First, we introduce Markov partitions for iterated function systems and give an algorithm that constructs a finite Markov partition whenever one exists. Second, for contractive systems admitting a finite Markov partition, we construct an intrinsic analogue of the Hutchinson measure that is independent of any labelling or weighting of the maps. It is the unique Borel probability measure satisfying a self-similar integral equation dictated by the set-valued dynamics of the system, and it recovers the Hutchinson measure with uniform weights when the system satisfies the open set condition. Finally, we give a complete classification of KMS states for the gauge action on the Kajiwara-Watatani algebra of every such system. The critical inverse temperature is the exponential growth rate of the intrinsic orbits, which can be strictly smaller than the natural logarithm of the number of maps, and the unique KMS state at this temperature is given by our intrinsic Hutchinson measure. These systems include many with large overlaps not covered by the results of Izumi-Kajiwara-Watatani.

A KMS-symmetric quantum Markov semigroup is completely determined by the quadratic form in induces on the GNS Hilbert space. The quadratic forms that arise in this way are called quantum Dirichlet forms. In this article we give a complete description of bounded quantum Dirichlet forms in terms of derivations. In particular, we exhibit an explicit characterization of the bounded quantum Dirichlet forms on type I factors. We also show that any unbounded quantum Dirichlet form can be represented in terms of derivations. The key technical insight is that the modular group always restricts to a strongly continuous group on the domain of a quantum Dirichlet form such that the square root of the analytic generator is accretive with respect to the form norm.

We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.

We prove that every finitely generated group with rapid decay and trivial amenable radical is C*-simple.

In this paper, we first introduce fiberwise supramenability for locally compact Hausdorff étale groupoids with compact unit space. Then, for such a minimal ample groupoid $\mathcal{G}$, we show that if $\mathcal{G}$ is fiberwise supramenable or topologically amenable, then the clopen type semigroup $S(\mathcal{G})$ is almost unperforated. Consequently, if $\mathcal{G}$ is $\sigma$-compact, then it has groupoid strict comparison. We then present several applications, including Matui's AH conjecture and pure infiniteness for groupoids and groupoid $C^*$-algebras.

We study the quantum Cayley graphs associated with the Kac-Paljutkin quantum group. We determine which projections give rise to undirected, irreflexive and connected quantum graphs and classify them up to isomorphism. Their spectra and chromatic numbers are computed. For the special case of the projection $p = I_2$ we also compute the associated quantum Cuntz-Krieger algebra.

We study C*-algebras generated by representations of product systems over unital subsemigroups of amenable discrete groups. In the first part, we show that exactness of the generated C*-algebra is equivalent to exactness of the cores over the constructible ideals. Nuclearity of the generated C*-algebra is equivalent to nuclear embeddability of the cores in the fixed point algebra. In the second part, we further study injective equivariant Fock covariant representations, and whether the properties of the cores are inherited from properties of the coefficient C*-algebra, as in the case of compactly aligned saturated product systems over right LCM semigroups. If the semigroup is finitely aligned and the product system is saturated and compactly aligned, then exactness of the cores is equivalent to exactness of the coefficient C*-algebra. If in addition the semigroup is strongly finitely aligned, then nuclear embeddability of the cores is equivalent to nuclear embeddability of the coefficient C*-algebra. These results do not hold without the compact alignment hypothesis, even for right LCM semigroups.

We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra $\mathcal O_\infty$. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of $\mathcal O_\infty$: one family being in an obvious bijection with the unit sphere of Hilbert's space.

We extend the generalised Calderon Transfer Principle as presented in \cite{dBL} to the setting of trace preserving group actions of $\sigma$-compact locally compact Hausdorff groups on semifinite von Neumann algebras $\M$. At its essence the transfer principle consists of showing that a large class of regular operators in the group context may be translated to operators in the algebra context by means of this group action. The end result is a protocol for proving ergodic convergence results for group actions on noncommutative Orlicz space of semifinite von Neumann algebras. We start with proving the existence of the transferred operator in the noncommutative context. Two tools are developed for the purpose of actually proving convergence results: (1) a theory of what may be called 2-variable decreasing rearrangements for the algebra $L^\infty \overline{\otimes} \M$ (where ($\Gamma,\nu$) is a Radon measure space) and (2) a concepts of maximal operators suited to the present context. These tools are then used to lift the ergodic convergence results presented in \cite{dBL} to the noncommutative setting. In closing we present examples illustrating the application of the tools

Kubo-Ando means are a rich class of important binary operations on the set of positive elements of an operator algebra. The aim of this article is to understand the algebraic structure of these operations. We generalize a recent result of L. Molnár for $C^*$-algebras to the class of split $C^*$-by-nilpotent algebras by using the method of analytic local expansion of binary operations. Eventually, we prove that in any $C^*$-algebra $\mathcal{A}$, the identity $[x,[y,z]]=0$ implies commutativity.

For a Dirac operator $D$ on a finite weighted graph, let $d^D$ denote the associated noncommutative (Connes) distance. We show that deleting an edge can actually decrease the noncommutative distance between two vertices in the graph. The smallest example of this phenomenon comes from a weighted 4-cycle, and the case of deleting an edge from a weighted 4-cycle is determined completely: if $d^{D'}$ denotes the noncommutative distance in the graph after edge deletion, we prove that $\sup d^D/d^{D'} = 2/\sqrt{3}$ for the $4$-cycle, and further that this upper bound holds for all graphs. On the other hand, we show that deleting an edge can decrease the noncommutative distance between two vertices of an $n$-cycle precisely when $n$ is divisible by $4$. In cases when a deleted edge decreases the noncommutative distance between two vertices $x$ and $y$, we show that the deleted edge need not be incident to either of the vertices $x$ or $y$, and in fact the deleted edge can be arbitrarily far from both in the graph-theoretic (number of edges) distance and in the weighted (geodesic) distance. Moreover, deletion of a single vertex arbitrarily far from both $x$ and $y$ in either the graph-theoretic or weighted distance can change $d^D(x,y)$ by an arbitrarily large factor.

Noncommutative polynomials, elements of the free algebra generated by several noncommuting variables $x_1,...,x_g,x_1^*,...,x_g^*$, are of importance in noncommutative algebra, free analysis and real algebraic geometry. We allow matrix coefficients and view these noncommutative polynomials as functions in several size-free matrix variables. The free locus of a noncommutative polynomial $f$ is the sequence of hypersurfaces $\mathscr{Z}_n(f)=\{X\in M_n(\mathbb{C})^g:\det f(X) = 0\}$. When $f$ is Hermitian, the positivity domain of $f$ is the sequence of semialgebraic sets $\mathcal{D}_n(f)=\{X\in M_n(\mathbb{C})^g:f(X)\text{ is positive semidefinite}\}$. We show for well-behaved cases that the Euclidean boundary of the positivity domain is Zariski dense in the free locus. This leads to a certificate for equality of positivity domains of coefficient-irreducible unsignatured atomic quadratics. Finally, we describe the geometric structure of positivity domains of hereditary quadratic polynomials through $\Gamma$-convexity.

We first solve in ZFC a problem that the first author and Weaver posed ten years ago: a state on a von Neumann algebra is countably additive on orthogonal projections if and only if it is sequentially weak* continuous, equivalently, sequentially normal. Such states may be considered the quantum (countably additive) probability measures, or rather their extension to a noncommutative integral. The principal tools for this are known ideas from direct integral theory from the 1970s. Indeed we prove a disintegration theorem for states on products of sigma-finite von Neumann algebras. A decomposition adapted to a given sequence reduces the continuity assertion in our main result to the classical dominated convergence theorem. The argument needs no separability hypothesis. We give several applications. For example we give a new variant of Gleason theorem, describing the countably additive projection measures for any von Neumann algebra with no type $I_2$ direct summand. We then turn to the weight case, discussing sequentially normal weights and the famous related Haagerup's Problem 1.11, giving some partial results (which may conceivably may turn out to be best possible in a certain sense). For example we solve Haagerup's problem if our weight is strongly or strictly semifinite, or with no restrictions for certain classes of von Neumann algebras. We give several applications of our main result to weights. In forthcoming work we consider many applications to `quantum measure and integration theory', such as variants of Lebesgue's dominated convergence theorem for von Neumann algebras.

For simple, nonelementary C*-algebras, we show uniform density of functions realized as ranks of operators. It follows that strict comparison by traces forces almost divisibility, and hence pureness. As an application, we show that strict comparison implies $\mathcal{Z}$-stability for separable, simple, nonelementary C*-algebras with locally finite nuclear dimension. In particular, the Toms–Winter conjecture holds for approximately subhomogeneous C*-algebras. More generally, we obtain uniform density of ranks for C*-algebras with the Global Glimm Property. Consequently, an exact C*-algebra is pure if and only if it has strict comparison together with the Global Glimm Property.

We answer, in the principal setting, a question of Matui on the almost finite–purely infinite dichotomy for minimal amenable ample groupoids. We prove that a second-countable Hausdorff minimal principal topologically amenable ample groupoid with Cantor unit space is strongly almost finite if and only if it admits an invariant probability measure; if no such measure exists, then it is purely infinite. We also prove that every bounded-degree uniformly Borel amenable Følner graph is Borel almost finite. The proofs combine the recent three-to-two comparison method of Glasner and Liu with the randomized Følner packing methods of Elek and Timár.

This survey studies C*-algebraic crossed products arising from topological dynamical systems with an eye toward their classifiability in the sense of the Elliott program. We introduce the crossed product construction for actions by discrete groups in full generality, and then focus on commutative systems to establish some of the fundamental structural results: nuclearity of the crossed product is equivalent to amenability of the action, and in this setting simplicity is equivalent to the combination of minimality and topological freeness. With these properties in place, the only condition left to translate into dynamical terms is tensorial absorption of the Jiang-Su algebra $\mathcal{Z}$. We present the main dynamical tools known to obtain $\mathcal{Z}$-stability, both in the settings of amenable and nonamenable groups, and highlight the two conjectures that are believed to capture the full picture. The aim of the survey is to provide both an accessible entry point and a comprehensive reference on the classification of crossed products, describing the state of the art and a number of central open problems in the field.