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Mehrdad Kalantar

Publications and source records attributed to Mehrdad Kalantar.

At least 19 recordsLinked to original sources
arXiv2026

Stabilizer Subgroups and the Simplicity of Reduced Crossed Products

Given a minimal action $G\curvearrowright X$ of a countable group $G$ on a compact space $X$, we prove that if the reduced crossed product $G\ltimes_rC(X)$ is simple, then there exists a point whose stabilizer subgroup has trivial amenable radical. As a consequence, we give a complete characterization of the simplicity of the reduced crossed product of minimal actions of countable linear groups, hyperbolic groups, and, more generally, for groups with countably many amenable subgroups. This answers a question of Ozawa (2014) for these classes of groups. Furthermore, in the case of an infinite uniformly recurrent subgroup of a $C^*$-simple group, we prove that almost every subgroup has a trivial amenable radical, with respect to a fully supported, atomless probability measure.

math.OA↗
arXiv2026

Growth conditions for topological freeness

Given a finitely generated group $\Gamma$, a non-trivial element $g\in \Gamma$, and a minimal action $\Gamma\curvearrowright \mathcal{X}$ on a compact space $\mathcal{X}$, with amenable neighborhood stabilizers, we prove sufficient conditions in terms of various growth/decay functions for topological freeness of the action of $g$ on $\mathcal{X}$. We apply our results to the case of the Furstenberg boundary action of $\Gamma$, to conclude C*-simplicity under (sub-)rapid decay conditions. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of $\Gamma$.

math.OA↗
arXiv2026

Conjugacy co-amenability

In this note we study a natural analytic property of inclusions of groups akin to co-amenability: the property of existence of a non-compactly supported invariant state for the conjugation action of a group $G$ on the von Neumann algebra generated by the characteristic functions $\{\mathbf{1}_{gHg^{-1}}\}_{g\in G}$ viewed inside $\ell^\infty(G)$. Some interesting settings and examples of this phenomena are proved. We also comment on a consideration related to proper proximality, which motivated this property.

math.OA↗
arXiv2025

Retraction Theorems for Group Compactifications

We characterize group compactifications of discrete groups for which there exists an equivariant retraction onto the boundary. In particular, we prove an equivariant analogue of Brouwer's No-Retraction theorem for large classes of group compactifications, which includes actions of hyperbolic groups on their Gromov boundary.

math.GR↗
arXiv2023

Operator space complexification transfigured

Given a finite group G, a central subgroup H of G, and an operator space X equipped with an action of H by complete isometries, we construct an operator space $X_G$ equipped with an action of G which is unique under a `reasonable' condition. This generalizes the operator space complexification $X_c$ of $X$. As a linear space $X_G$ is the space obtained from inducing the representation of H to G (in the sense of Frobenius).

math.OA↗
arXiv2023

Real structure in operator spaces, injective envelopes and $G$-spaces

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and $C^*$-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.

math.OA↗
arXiv2022

Boundary maps and covariant representations

We extend applications of Furstenberg boundary theory to the study of $C^*$-algebras associated to minimal actions $Γ\!\curvearrowright\! X$ of discrete groups $Γ$ on locally compact spaces $X$. We introduce boundary maps on $(Γ,X)$-$C^*$-algebras and investigate their applications in this context. Among other results, we completely determine when $C^*$-algebras generated by covariant representations arising from stabilizer subgroups are simple. We also characterize the intersection property of locally compact $Γ$-spaces and simplicity of their associated crossed products.

math.OA↗
arXiv2022

A type I conjecture and boundary representations of hyperbolic groups

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group $G$ associated with non-singular $G$-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Ł. Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

math.GR↗
arXiv2022

On invariant subalgebras of group $C^*$ and von Neumann algebras

Given an irreducible lattice $Γ$ in the product of higher rank simple Lie groups, we prove a co-finiteness result for the $Γ$-invariant von Neumann subalgebras of the group von Neumann algebra $\mathcal{L}(Γ)$, and for the $Γ$-invariant unital $C^*$-subalgebras of the reduced group $C^*$-algebra $C^*_{\rm red}(Γ)$. We use these results to show that: (i) every $Γ$-invariant von Neumann subalgebra of $\mathcal{L}(Γ)$ is generated by a normal subgroup; and (ii) given a non-amenable unitary representation $π$ of $Γ$, every $Γ$-equivariant conditional expectation on $C^*_π(Γ)$ is the canonical conditional expectation onto the $C^*$-subalgebra generated by a normal subgroup.

math.OA↗
arXiv2021

Boundary maps, germs and quasi-regular representations

We investigate the tracial and ideal structures of $C^*$-algebras of quasi-regular representations of stabilizers of boundary actions. Our main tool is the notion of boundary maps, namely $Γ$-equivariant unital completely positive maps from $Γ$-$C^*$-algebras to $C(\partial_FΓ)$, where $\partial_FΓ$ denotes the Furstenberg boundary of a group $Γ$. For a unitary representation $π$ coming from the groupoid of germs of a boundary action, we show that there is a unique boundary map on $C^*_π(Γ)$. Consequently, we completely describe the tracial structure of the $C^*$-algebras $C^*_π(Γ)$, and for any $Γ$-boundary $X$, we completely characterize the simplicity of the $C^*$-algebras generated by the quasi-regular representations $λ_{Γ/Γ_x}$ associated to stabilizer subgroups $Γ_x$ for any $x\in X$. As an application, we show that the $C^*$-algebra generated by the quasi-regular representation $λ_{T/F}$ associated to Thompson's groups $F\leq T$ does not admit traces and is simple.

math.OA↗
arXiv2021

Tight inclusions of C*-dynamical systems

We study a notion of tight inclusions of C*- and W*-dynamical systems which is meant to capture a tension between topological and measurable rigidity of boundary actions. An important case of such inclusions are $C(X)\subset L^\infty(X, ν)$ for measurable boundaries with unique stationary compact models. We discuss the implications of this phenomenon in the description of Zimmer amenable intermediate factors. Furthermore, we prove applications in the problem of maximal injectivity of von Neumann algebras.

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arXiv2020

Stationary C*-dynamical systems

We introduce the notion of stationary actions in the context of C*-algebras. We develop the basics of the theory, and provide applications to several ergodic theoretical and operator algebraic rigidity problems.

math.OA↗
arXiv2020

Noncommutative Furstenberg boundary

We introduce and study the notions of boundary actions and of the Furstenberg boundary of a discrete quantum group. As for classical groups, properties of boundary actions turn out to encode significant properties of the operator algebras associated with the discrete quantum group in question; for example we prove that if the action on the Furstenberg boundary is faithful, the quantum group C*-algebra admits at most one KMS-state for the scaling automorphism group. To obtain these results we develop a version of Hamana's theory of injective envelopes for quantum group actions, and establish several facts on relative amenability for quantum subgroups. We then show that the Gromov boundary actions of free orthogonal quantum groups, as studied by Vaes and Vergnioux, are also boundary actions in our sense; we obtain this by proving that these actions admit unique stationary states. Moreover, we prove these actions are faithful, hence conclude a new unique KMS-state property in the general case, and a new proof of unique trace property when restricted to the unimodular case. We prove equivalence of simplicity of the crossed products of all boundary actions of a given discrete quantum group, and use it to obtain a new simplicity result for the crossed product of the Gromov boundary actions of free orthogonal quantum groups.

math.OA↗
arXiv2020

On fixed point property for $L_p$-representations of Kazhdan groups

Let $G$ be a topological group with finite Kazhdan set, let $Ω$ be a standard Borel space and $μ$ a finite measure on $Ω$. We prove that for any $p\in [1, \infty)$, any affine isometric action $G \curvearrowright L_p(Ω, μ)$ whose linear part arises from an ergodic measure-preserving action $G \curvearrowright (Ω, μ)$, has a fixed point.

math.GR↗
arXiv2019

Topological boundaries of unitary representations

We introduce and study a generalization of the notion of the Furstenberg boundary of a discrete group $Γ$ to the setting of a general unitary representation $π: Γ\to B(\mathcal H_π)$. This space, which we call the "Furstenberg-Hamana boundary" of the pair $(Γ, π)$, is a $Γ$-invariant subspace of $B(\mathcal H_π)$ that carries a canonical $C^*$-algebra structure. In many natural cases, including when $π$ is a quasi-regular representation, the Furstenberg-Hamana boundary of $π$ is commutative, but can be non-commutative in general. We study various properties of this boundary, and give some applications.

math.OA↗
arXiv2019

Quasi-regular representations of discrete groups and associated C*-algebras

Let $G$ be a countable group. We introduce several equivalence relations on the set ${\rm Sub}(G)$ of subgroups of $G$, defined by properties of the quasi-regular representations $λ_{G/H}$ associated to $H\in {\rm Sub}(G)$ and compare them to the relation of $G$-conjugacy of subgroups. We define a class ${\rm Sub}_{\rm sg}(G)$ of subgroups (these are subgroups with a certain spectral gap property) and show that they are rigid, in the sense that the equivalence class of $H\in {\rm Sub}_{\rm sg}(G)$ for any one of the above equivalence relations coincides with the $G$-conjugacy class of $H$. Next, we introduce a second class ${\rm Sub}_{\rm w-par}(G)$ of subgroups (these are subgroups which are weakly parabolic in some sense) and we establish results concerning the ideal structure of the $C^*$-algebra $C^*_{λ_{G/H}}(G)$ generated by $λ_{G/H}$ for subgroups $H$ which belong to either one of the classes ${\rm Sub}_{\rm w-par}(G)$ and ${\rm Sub}_{\rm sg}(G)$. Our results are valid, more generally, for induced representations ${\rm Ind}_H^G σ$, where $σ$ is a representation of $H\in {\rm Sub}(G)$.

math.GR↗