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Mehrdad Kalantar

Publications and source records attributed to Mehrdad Kalantar.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2018

On simplicity of intermediate C*-algebras

We prove simplicity of all intermediate $C^*$-algebras $C^*_{r}(Γ)\subseteq \mathcal{B} \subseteq Γ\ltimes_r C(X)$ in the case of minimal actions of $C^*$-simple groups $Γ$ on compact spaces $X$. For this, we use the notion of stationary states, recently introduced by Hartman and Kalantar (arXiv:1712.10133v2). We show that the Powers' averaging property holds for the reduced crossed product $Γ\ltimes_r \mathcal{A}$ for any action $Γ\curvearrowright \mathcal{A}$ of a $C^*$-simple group $Γ$ on a unital $C^*$-algebra $\mathcal{A}$, and use it to prove a one-to-one correspondence between stationary states on $\mathcal{A}$ and those on $Γ\ltimes_r \mathcal{A}$.

math.OA↗
arXiv2018

Rokhlin dimension for compact quantum group actions

We show that, for a given compact or discrete quantum group $G$, the class of actions of $G$ on C*-algebras is first-order axiomatizable in the logic for metric structures. As an application, we extend the notion of Rokhlin property for $G$-C*-algebra, introduced by Barlak, Szabó, and Voigt in the case when $G$ is second countable and coexact, to an arbitrary compact quantum group $G$. All the the preservations and rigidity results for Rokhlin actions of second countable coexact compact quantum groups obtained by Barlak, Szabó, and Voigt are shown to hold in this general context. As a further application, we extend the notion of equivariant order zero dimension for equivariant *-homomorphisms, introduced in the classical setting by the first and third authors, to actions of compact quantum groups. This allows us to define the Rokhlin dimension of an action of a compact quantum group on a C*-algebra, recovering the Rokhlin property as Rokhlin dimension zero. We conclude by establishing a preservation result for finite nuclear dimension and finite decomposition rank when passing to fixed point algebras and crossed products by compact quantum group actions with finite Rokhlin dimension.

math.OA↗
arXiv2017

Induction for locally compact quantum groups revisited

In this paper we revisit the theory of induced representations in the setting of locally compact quantum groups. In the case of induction from open quantum subgroups, we show that constructions of Kustermans and Vaes are equivalent to the classical, and much simpler, construction of Rieffel. We also prove in general setting the continuity of induction in the sense of Vaes with respect to weak containment.

math.OA↗
arXiv2016

C*-simplicity and the unique trace property for discrete groups

A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.

math.OA↗
arXiv2016

Open quantum subgroups of locally compact quantum groups

The notion of an open quantum subgroup of a locally compact quantum group is introduced and given several equivalent characterizations in terms of group-like projections, inclusions of quantum group C*-algebras and properties of respective quantum homogenous spaces. Open quantum subgroups are shown to be closed in the sense of Vaes and normal open quantum subgroups are proved to be in 1-1 correspondence with normal compact quantum subgroups of the dual quantum group.

math.OA↗
arXiv2015

The classification problem for finitely generated operator systems and spaces

The classification of separable operator spaces and systems is commonly believed to be intractable. We analyze this belief from the point of view of Borel complexity theory. On one hand we confirm that the classification problems for arbitrary separable operator systems and spaces are intractable. On the other hand we show that the finitely generated operator systems and spaces are completely classifiable (or smooth); in fact a finitely generated operator system is classified by its complete theory when regarded as a structure in continuous logic. In the particular case of operator systems generated by a single unitary, a complete invariant is given by the spectrum of the unitary up to a rigid motion of the circle, provided that the spectrum contains at least 5 points. As a consequence of these results we show that the relation on compact subsets of $\mathbb{C}^{n}$, given by homeomorphism via a degree 1 polynomial, is smooth.

math.OA↗
arXiv2014

An uncertainty principle for unimodular quantum groups

We present a generalization of Hirschman's entropic uncertainty principle for locally compact abelian groups to unimodular locally compact quantum groups. As a corollary, we strengthen a well-known uncertainty principle for compact groups, and generalize the relation to compact quantum groups of Kac type. We also establish the complementarity of finite-dimensional quantum group algebras. In the non-unimodular setting, we obtain an uncertainty relation for arbitrary locally compact groups using the relative entropy with respect to the Haar weight as the measure of uncertainty. We also show that when restricted to normal central states of discrete quantum groups, the relative entropy with respect to the Haar weight reduces to the canonical entropy of the random walk generated by the central state.

math-ph↗
arXiv2014

Boundaries of reduced C*-algebras of discrete groups

For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up to isomorphism. It is trivial if and only if G is amenable. We prove that, more generally, it can be identified with the algebra $C(\partial_F G)$ of continuous functions on Furstenberg's universal G-boundary $\partial_F G$. This operator-algebraic construction of the Furstenberg boundary has a number of interesting consequences. We prove that G is exact precisely when the G-action on $\partial_F G$ is amenable, and use this fact to prove Ozawa's conjecture that if G is exact, then there is an embedding of the reduced C*-algebra $\mathrm{C}_r^*(G)$ of G into a nuclear C*-algebra which is contained in the injective envelope of $\mathrm{C}_r^*(G)$. It is a longstanding open problem to determine which groups are C*-simple, in the sense that the algebra $\mathrm{C}_r^*(G)$ is simple. We prove that this problem can be reformulated as a problem about the structure of the G-action on the Furstenberg boundary. Specifically, we prove that a discrete group G is C*-simple if and only if the G-action on the Furstenberg boundary is topologically free. We apply this result to prove that Tarski monster groups are C*-simple. This provides another solution to a problem of de la Harpe (recently answered by Olshanskii and Osin) about the existence of C*-simple groups with no free subgroups.

math.OA↗
arXiv2014

A note on concentration functions on transformation groups

In this note, we consider the concentration function problem for a continuous action of a locally compact group $G$ on a locally compact Hausdorff space $X$. We prove a necessary and sufficient condition for the concentration functions of a spread-out irreducible probability measure $μ$ on $G$ to converge to zero.

math.DS↗
arXiv2014

On iterated powers of positive definite functions

We prove that if $ρ$ is an adapted positive definite function in the Fourier--Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\|ρ\|_{B(G)}=1$, then the iterated powers $(ρ^n)$ converge to zero in the weak* topology $σ(B(G) , C^*(G))$. Moreover, if $ρ$ is irreducible, we prove that $(ρ^n)$ as a sequence of u.c.p. maps on the group $C^*$-algebra converges to zero in the strong operator topology.

math.OA↗
arXiv2014

Realization of quantum group Poisson boundaries as crossed products

For a locally compact quantum group $\mathbb{G}$, consider the convolution action of a quantum probability measure $μ$ on $L_\infty(\mathbb{G})$. As shown by Junge--Neufang--Ruan, this action has a natural extension to a Markov map on $\mathcal{B}(L_2(\mathbb{G}))$. We prove that the Poisson boundary of the latter can be realized concretely as the von Neumann crossed product of the Poisson boundary associated with $μ$ under the action of $\mathbb{G}$ induced by the coproduct. This yields an affirmative answer, for general locally compact quantum groups, to a problem raised by Izumi (2004) in the commutative situation, in which he settled the discrete case, and unifies earlier results of Jaworski, Neufang and Runde.

math.OA↗
arXiv2014

Harmonic Operators of Ergodic Quantum Group Actions

In this paper we study the harmonic elements of (convolution) Markov maps associated to (ergodic) actions of locally compact quantum groups on ($σ$-finite) von Neumann algebras. We give several equivalent conditions under which the harmonic elements are trivial.

math.OA↗
arXiv2014

Poisson boundaries over locally compact quantum groups

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet--Deny theorem holds for compact quantum groups; also, the result of Kaimanovich--Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a non-commutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group $SU_{q}(2)$ arising from measures on its spectrum.

math.OA↗
arXiv2013

A Limit Theorem for Discrete Quantum Groups

We consider the {\it concentration functions problem} for discrete quantum groups; we prove that if $\mathbb{G}$ is a discrete quantum group, and $μ$ is an irreducible state in $l^1(\mathbb{G})$, then the convolution powers $μ^n$, considered as completely positive maps on $c_0(\mathbb{G})$, converge to zero in strong operator topology.

math.OA↗
arXiv2013

Compact Operators in Regular LCQ Groups

We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $L^\infty(\mathbb{G})$ contains non-zero compact operators on $L^2(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $L^1(\mathbb{G})$ has the Radon--Nikodym property.

math.OA↗