CyberSec Research

We continue with the theory of real noncommutative (nc) convexity, following the recent and profound complex case developed by Davidson and Kennedy. The present paper focuses on the theory of nc extreme (and pure and maximal) points and the nc Choquet boundary in the real case, and on the theory of real nc convex and semicontinuous functions and real nc convex envelopes. Our main emphasis is on how these interact with complexification. For example some of our paper analyzes carefully how various notions of `extreme' or `maximal' interact with our earlier concept of the complexification of a convex set. Several new features appear in the real case in later sections of our paper, including the novel notion of the complexification of a nc convex function, and the complexification of the convex envelope of a nc function. With an Appendix by T. Russell.

By Bartle-Graves theorem every surjective map between C*-algebras has a continuous section, and Loring proved that that there exists a continuous section of norm arbitrary close to 1. Here we prove that there exists a continuous section of norm exactly 1. This result is used in the second part of the paper which is devoted to properties of cone C*-algebras. It is proved that any $\ast$-homomorphism from the cone over a separable C*-algebra to a quotient C*-algebra always lifts to a contractive asymptotic homomorphism. As an application we give a short proof and strengthen the result of Forough-Gardella-Thomsen that states that any cpc (order zero) map has an asymptotically cpc (order zero, respectively) lift. As another application we give unified proofs of Voiculescu's result that cones are quasidiagonal and Brown-Carrion-White's result that all amenable traces on cones are quasidiagonal. We also prove that all hyperlinear traces on cones are MF.

This paper investigates derivations of the free semigroupoid algebra $\mathfrak{L}_G$ of a countable or uncountable directed graph $G$ and its norm-closed version, the tensor algebra $\mathcal{A}_G$. We first prove a weak Dixmier approximation theorem for $\mathfrak{L}_G$ when $G$ is strongly connected. Using the theorem, we show that if every connected component of $G$ is strongly connected, then every bounded derivation $\delta$ from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is of the form $\delta=\delta_T$ for some $T\in\mathfrak{L}_G$ with $\|T\|\leqslant\|\delta\|$. For any finite directed graph $G$, we also show that the first cohomology group $H^1(\mathcal{A}_G,\mathfrak{L}_G)$ vanishes if and only if every connected component of $G$ is either strongly connected or a fruit tree. To handle infinite directed graphs, we introduce the alternating number and propose \Cref{conj intro-in-tree}. Suppose every connected component of $G$ is not strongly connected. We show that if every bounded derivation from $\mathcal{A}_G$ into $\mathfrak{L}_G$ is inner, then every connected component of $G$ is a generalized fruit tree and the alternating number $A(G)$ of $G$ is finite. The converse is also true if the conjecture holds. Finally, we provide some examples of free semigroupoid algebras together with their nontrivial first cohomology groups.

We establish character rigidity for all non-uniform higher-rank irreducible lattices in semisimple groups of characteristic other than 2. This implies stabilizer rigidity for probability measure preserving actions and rigidity of invariant random subgroups, confirming a conjecture of Stuck and Zimmer for non-uniform lattices in full generality.

We characterize invariant subspaces of Brownian shifts on vector-valued Hardy spaces. We also solve the unitary equivalence problem for the invariant subspaces of these shifts.

We investigate isometric and algebraic isomorphism problems for multiplier algebras associated with Hilbert spaces of Dirichlet series whose kernels possess the complete Nevanlinna-Pick (CNP) property. We begin by providing a complete characterization of all normalized CNP Dirichlet series kernels in terms of weight and frequency data. A central aspect of our work is the explicit determination of the multiplier varieties associated with CNP Dirichlet series kernels. We show that these varieties are defined by explicit polynomial equations derived from the arithmetic structure of the weight and frequency data associated with the kernel. This explicit description of multiplier varieties enables us to classify when the multiplier algebras of certain CNP Dirichlet series kernels are (isometrically) isomorphic. As an application, we resolve an open question posed by McCarthy and Shalit ([18]) in the negative.

Using the Witten deformation and localization algebra techniques, we compute the $G$-equivariant $K$-homology class of the de Rham operator on a proper cocompact $G$-spin manifold, where $G$ is an almost connected Lie group. By applying a $G$-invariant Morse-Bott perturbation, this class is localized near the zero set of the perturbation and can be identified explicitly with an element in the representation rings associated to some isotropy subgroups. The result yields an equivariant Poincar\'e-Hopf formula and supplies concise tools for equivariant index computations.

Given two second order free random variables $a$ and $b$, we study the second order free cumulants of their product $ab$, their commutator $ab-ba$, and their anti-commutator $ab+ba$. Let $(\kappa_n^a)_{n\geq 1}$ and $(\kappa_{n,m}^a)_{n,m\geq 1}$ denote the sequence of free cumulants of first and second order, respectively, of a random variable $a$ in a second order non-commutative probability space $(\mathcal{A},\varphi,\varphi^2)$. Given $a$ and $b$ two second order freely independent random variables, we provide formulas to compute each of the cumulants $(\kappa_{n,m}^{ab})_{n,m\geq 1}$, $(\kappa_{n,m}^{ab-ba})_{n,m\geq 1}$, and $(\kappa_{n,m}^{ab+ba})_{n,m\geq 1}$ in terms of the individual cumulants $(\kappa_{n}^{a})_{n\geq 1}$, $(\kappa_{n,m}^{a})_{n,m\geq 1}$, $(\kappa_{n}^{b})_{n\geq 1}$, and $(\kappa_{n,m}^{b})_{n,m\geq 1}$. For $n=m=1$ our formulas read: \begin{align*} \kappa_{1,1}^{ab} &= \kappa_{2}^{a}\kappa_{2}^{b} +\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+\kappa_{1,1}^{b}(\kappa_{1}^{a})^2,\\ \kappa_{1,1}^{ab-ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b},\\ \kappa_{1,1}^{ab+ba} &= 2\kappa_{2}^{a}\kappa_{2}^{b} +4\kappa_{1,1}^{a}(\kappa_{1}^{b})^2+4\kappa_{1,1}^{b}(\kappa_{1}^{a})^2. \end{align*} In general, our formulas express the cumulants $\kappa_{n,m}^{ab}$, $\kappa_{n,m}^{ab-ba}$, and $\kappa_{n,m}^{ab+ba}$ as sums indexed by special subsets of non-crossing partitioned permutations. The formulas for the commutator and anti-commutator where not studied before, while the formula for the product was only known in the case the where the individual second order free cumulants vanish. As an application, we compute explicitly the cumulants of the anti-commutator and product of two second order free semicircular variables.

We analyze a natural C*-algebraic definition of G-quasi-invariant states for the automorphic action of a compact group G. We prove that, given a G-quasi-invariant state with central support, when the action of the group G commutes with the modular group, its GNS representation is equivalent to that of a G-invariant state.

The classical Choquet theorem establishes a barycentric decomposition for elements in a compact convex subset of a locally convex topological vector space. This decomposition is achieved through a probability measure that is supported on the set of extreme points of the subset. In this work, we consider a partial action $\tau$ of a group $G$ on a $C^\ast$-algebra $\mathcal{A}$. For a fixed Hilbert space $\mathcal{H}$, we consider the set of all unital completely positive maps from $\mathcal{A}$ to $\mathcal{B}(\mathcal{H})$ that are invariant under the partial action $\tau$. This set forms a compact convex subset of a locally convex topological vector space. To complete the picture of the barycentric decomposition provided by the classical Choquet theorem, we characterize the set of extreme points of this set.

Unitary vertex operator algebras (VOAs) and conformal nets are the two most prominent mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. They are conjectured to be equivalent, but a rigorous comparison has proven challenging. We resolve one direction of the conjecture by showing that every conformal net has an associated unitary VOA.

The main purpose of this article is to explore the possibility of extending the notion of peripheral Poisson boundary of unital completely positive (UCP) maps to contractive completely positive (CCP) maps and to unital and non-unital contractive quantum dynamical semigroups on von Neumann algebras. We observe that the theory extends easily in the setting of von Neumann algebras and normal maps. Surprisingly, the peripheral Poisson boundary is unital, whenever it is nontrivial, even for contractive semigroups. The strong operator limit formula for computing the extended Choi-Effros product remains intact. However, there are serious obstacles in the framework of $C^*$-algebras, and we are unable to define the extended Choi-Effros product in such generality. We provide several intriguing examples to illustrate this.

For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced $C^{*}$-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding $E_0$-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced $C^{*}$-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of $K$-theory under homotopy of product systems.

For non-amenable finitely generated virtually free groups, we show that the combinatorial Euler characteristic introduced by Emerson and Meyer is the preimage of the K-theory class of higher Kazhdan projections under the Baum-Connes assembly map. This allows to represent the K-theory class of their higher Kazhdan projection as a finite alternating sum of the K-theory classes of certain averaging projections. The latter is associated to the finite subgroups appearing in the fundamental domain of their Bass-Serre tree. As an immediate application we obtain non-vanishing calculations for delocalised $\ell^2$-Betti numbers for this class of groups.

This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph $\Lambda$ without sources, we show that there exists a $\mathbb{Z}[\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\mathcal{G}_\Lambda)$ of the infinite path groupoid $\mathcal{G}_\Lambda$ and the graded Grothendieck group $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$, which respects the positive cones (i.e., the talented monoids). We demonstrate that the $k$-graph moves of in-splitting and sink deletion defined by Eckhardt et al. (Canad. J. Math. 2022) preserve the graded $K$-theory of associated Kumjian-Pask algebras and produce algebras which are graded Morita equivalent, thus providing evidence that graded $K$-theory may be an effective invariant for classifying certain Kumjian-Pask algebras. We also determine a natural sufficient condition regarding the fullness of the graded Grothendieck group functor. More precisely, for two row-finite $k$-graphs $\Lambda$ and $\Omega$ without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving $\mathbb{Z}[\mathbb{Z}^k]$-module homomorphism between $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ and $K_0^{gr}(KP_\mathsf{k}(\Omega))$ to a unital graded ring homomorphism between $KP_\mathsf{k}(\Lambda)$ and $KP_\mathsf{k}(\Omega)$. For this we adapt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).

It is proved that for every $r,s \in \mathbb{N}\backslash \{0,1\}$ the action of $\mathbb{F}_{r+s}$ on $\partial_\beta (\mathbb{F}_{r+s}/\mathbb{F}_r)$ is topologically amenable. In particular the $C^*$-algebra associated to the corresponding quasi-regular representation has a unique non-trivial ideal. The techniques involved rely on a study of dynamical properties for actions on non-standard boundaries studied by the author and F. R{\u a}dulescu in previous works.

In this paper we leverage the recently developed theory of noncommutative (nc) measures to prove a free noncommutative analogue of many known equalities extending the weak Szeg\H{o} limit theorem, by applying Constantinescu's theory of Schur parameters to an appropriate kernel on the free monoid on $d$ generators, where $d \geq 1$; in particular, we show that our nc Szeg\H{o} entropy depends only upon the absolutely continuous part of the associated nc measure. We obtain a correspondence between nc measures and multi-Toeplitz kernels arising from considering the moments of the nc measure, and apply this correspondence to study orthogonal polynomials associated to an nc measure. Finally, we study the determinantal zeros of those polynomials and obtain a noncommutative row-ball analogue of the so-called Zeros Theorem for orthogonal polynomials on the unit circle.

Finite Similarity Structure (FSS) groups are a class of generalized Thompson groups first introduced by Farley and Hughes. In this paper, we study the properties of a new subclass of the more general Countable Similarity Structure (CSS) groups, which we call CSS$^*$ groups. The Higman-Thompson groups $V_{d,r}$, the countable R\"over-Nekrashevych groups $V_d(G)$, and the topological full groups of irreducible one-sided subshifts of finite type studied by Matui are all examples of CSS$^*$ groups. One overarching theme is to isolate a class of CSS groups with the necessary properties to generalize the main result of arXiv:2312.08345 -- primeness of the group von Neumann algebra of $V_d$. We achieve this aim for CSS$^*$ groups and in the process prove that CSS$^*$ groups are non-inner amenable and many are properly proximal, which are new results for $V_{d,r}$, $V_d(G)$, and the groups studied by Matui. A second theme is to produce a dichotomy within CSS$^*$ groups; those that are $C^*$-simple and possess a simple commutator subgroup, and those lacking both properties. In particular, we extend the $C^*$-simplicity results of $V$ and $V(G)$ of arXiv:1605.01651, recover the simple commutator subgroup results of arXiv:2008.04791 and arXiv:1210.5800, and give examples that lack both properties. Lastly, we observe that CSS$^*$ groups are not acylindrically hyperbolic, motivating the need to prove many of these results by other methods.