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Mario Klisse

Publications and source records attributed to Mario Klisse.

14 recordsLinked to original sources
arXiv2026

Pureness and stable rank one for reduced twisted group $\mathrm{C}^\ast$-algebras of certain group extensions

We present a sufficient condition for a (twisted) group action on a $\mathrm{C}^\ast$-algebra such that the corresponding reduced (twisted) group $\mathrm{C}^\ast$-algebra inclusion into the reduced (twisted) crossed product is selfless. This condition is an action-dependent version of Ozawa's $\mathrm{PHP}$ property for groups. We show that topologically free extreme boundary actions, projective actions of lattices in ${\rm PSL}(n\ge 2,\mathbb R)$, as well as certain strongly proximal actions have this property, and thus we provide examples of selfless inclusions from crossed products. As a byproduct, we also obtain that reduced (twisted) group $\mathrm{C}^\ast$-algebras of some group extensions of the form finite-by-$G$, with $G$ having the property $\mathrm{PHP}$, have stable rank one and are pure, and in particular have strict comparison. Examples include all acylindrically hyperbolic groups and all lattices in ${\rm SL}(n,\mathbb R)$ for $n\geq2$.

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arXiv2026

Word-Length Spectral Triples of $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$ Are Not Metric

Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.

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arXiv2026

Selfless reduced free products and graph products of $\mathrm{C}^\ast$-algebras

Under mild assumptions, we show that reduced free products and reduced graph products of $\mathrm{C}^{\ast}$-algebras are completely selfless, without assuming the rapid decay property. In particular, our main theorems yield numerous new examples of simple, monotracial $\mathrm{C}^{\ast}$-algebras with strict comparison, stable rank one, and admitting a unique unital embedding of the Jiang--Su algebra $\mathcal{Z}$ up to approximate unitary equivalence, and of purely infinite $\mathrm{C}^{\ast}$-algebras in the traceless case.

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arXiv2025

Quantum Metric Structures on Iwahori-Hecke Algebras

Iwahori-Hecke algebras are $q$-deformations of group algebras of Coxeter groups. In this article, we initiate a systematic study of quantum metric structures on Iwahori-Hecke algebras by establishing that, for finite rank right-angled Coxeter systems, the canonical filtrations of the corresponding Iwahori-Hecke algebras satisfy the Haagerup-type condition introduced by Ozawa and Rieffel if and only if the Coxeter diagram's complement contains no induced squares. As a consequence, these algebras naturally inherit compact quantum metric space structures in the sense of Rieffel. Additionally, we investigate continuity phenomena in this framework by demonstrating that, as the deformation parameter $q$ approaches $1$, the deformed Iwahori-Hecke algebras converge to the group algebra of the Coxeter group in Latrémolière's quantum Gromov-Hausdorff propinquity.

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arXiv2025

Universal C$^{\ast}$-Algebras from Graph Products: Structure and Applications

In this article, we introduce and investigate a class of C$^{\ast}$-algebras generated by reduced graph products of C$^{\ast}$-algebras, augmented with families of projections naturally associated with words in right-angled Coxeter groups. These ambient C$^{\ast}$-algebras possess a rich and tractable combinatorial structure, which enables the deduction of a variety of structural properties. Among other results, we establish universal properties, characterize nuclearity and exactness in terms of the vertex algebras, and analyze the ideal structure. In the second part of the article, we leverage this framework to derive new insights into the structure of graph product C$^{\ast}$-algebras -- many of which are novel even in the case of free products.

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arXiv2025

Complementation in Continuous Cohomology with Coefficients in Banach Modules

In this article, we introduce the concept of weakly uniquely stationary representations. This framework enables us to investigate the complementability of closed subspaces within the context of continuous cohomology with coeffcients in Banach modules. As an application, we extend and refine several cohomological results from the literature, particularly in the settings of nilpotent groups, products of groups, and lattices.

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arXiv2024

Crossed products as compact quantum metric spaces

By employing the external Kasparov product, Hawkins, Skalski, White and Zacharias constructed spectral triples on crossed product C$^\ast$-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional non-commutative tori.

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arXiv2024

The Choquet-Deny Property for Groupoids

A countable discrete group is called Choquet-Deny if for any non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Building on the previous work of Jaworski, a complete characterization of Choquet-Deny groups was recently achieved by Frisch, Hartman, Tamuz, and Ferdowski. In this article, we extend the study of the Choquet-Deny property to the framework of discrete measured groupoids. Our primary result offers a complete characterization of this property in terms of the isotropy groups and the equivalence relation associated with the given groupoid. Additionally, we use the implications derived from our main theorem to classify the Choquet-Deny property of transformation groupoids.

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arXiv2023

Relative Haagerup property for arbitrary von Neumann algebras

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

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arXiv2022

Simplicity of right-angled Hecke C$^\ast$-algebras

By exploiting properties of boundaries associated with Coxeter groups we obtain a complete characterization of simple right-angled multi-parameter Hecke C$^{\ast}$-algebras. This extends previous results by Caspers, Larsen and the author. Based on work by Raum and Skalski, we further describe the center and the character space of right-angled Hecke C$^{\ast}$-algebras.

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arXiv2022

On the isomorphism class of $q$-Gaussian C$^\ast$-algebras for infinite variables

For a real Hilbert space $H_{\mathbb{R}}$ and $-1 < q < 1$ Bozejko and Speicher introduced the C$^\ast$-algebra $A_q(H_{\mathbb{R}})$ and von Neumann algebra $M_q(H_{\mathbb{R}})$ of $q$-Gaussian variables. We prove that if $\dim(H_{\mathbb{R}}) = \infty$ and $-1 < q < 1, q \not = 0$ then $M_q(H_{\mathbb{R}})$ does not have the Akemann-Ostrand property with respect to $A_q(H_{\mathbb{R}})$. It follows that $A_q(H_{\mathbb{R}})$ is not isomorphic to $A_0(H_{\mathbb{R}})$. This gives an answer to the C$^\ast$-algebraic part of Question 1.1 and Question 1.2 in [NeZe18].

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arXiv2021

Topological boundaries of connected graphs and Coxeter groups

We introduce and study certain topological spaces associated with connected rooted graphs. These spaces reflect combinatorial and order theoretic properties of the underlying graph and relate in the case of hyperbolic graphs to Gromov's hyperbolic compactification. They are particularly tractable in the case of Cayley graphs of finite rank Coxeter groups. In that context we speak of the compactification and the boundary of the Coxeter group. As it turns out, the canonical action of the Coxeter group on its Cayley graph induces a natural action on the compactification and the boundary. From this we deduce that in this case our construction coincides with spaces defined Caprace and Lécureux. We further prove the amenability of the action, we characterize when the compactification is small at infinity and we study classes of Coxeter groups for which the action is a topological boundary action in the sense of Furstenberg. The second part of the paper deals with the applications of our results to the study of (Iwahori) Hecke algebras. These are certain deformations of group algebras of Coxeter groups. We first study embeddings of Hecke C$^\ast$-algebras and prove property Akemann-Ostrand for a certain class of Hecke-von Neumann algebras. Lastly, we make use of results that are widely related to Kalantar-Kennedy's approach to the C$^\ast$-simplicity problem to study the simplicity and injective envelopes of operator algebras associated with Hecke algebras.

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arXiv2020

Graph product Khintchine inequalities and Hecke C*-algebras: Haagerup inequalities, (non)simplicity, nuclearity and exactness

Graph products of groups were introduced by Green in her thesis. They have an operator algebraic counterpart introduced and explored by Fima and the first-named author. In this paper we prove Khintchine type inequalities for general C$^{\ast}$-algebraic graph products which generalize results by Ricard and Xu on free products of C$^{\ast}$-algebras. We apply these inequalities in the context of (right-angled) Hecke C$^{\ast}$-algebras, which are deformations of the group algebra of Coxeter groups. For these we deduce a Haagerup inequality. We further use this to study the simplicity and trace uniqueness of (right-angled) Hecke C$^{\ast}$-algebras. Lastly we characterize exactness and nuclearity of general Hecke C$^{\ast}$-algebras.

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