arXiv subjects

Sven Raum

Publications and source records attributed to Sven Raum.

34 records · Page 2Linked to original sources

arXiv2017

L^2-Betti numbers of rigid C*-tensor categories and discrete quantum groups

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group $G$ are equal to the $L^2$-Betti numbers of the representation category $Rep(G)$ and thus, in particular, invariant under monoidal equivalence. As an application, we obtain several new computations of $L^2$-Betti numbers for discrete quantum groups, including the quantum permutation groups and the free wreath product groups. Finally, we obtain upper bounds for the first $L^2$-Betti number in terms of a generating set of a $C^*$-tensor category.

math.OA↗

arXiv2017

On the l^2-Betti numbers of universal quantum groups

We show that the first $\ell^2$-Betti number of the duals of the free unitary quantum groups is one, and that all $\ell^2$-Betti numbers vanish for the duals of the quantum automorphism groups of full matrix algebras.

math.OA↗

arXiv2016

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.

math.GR↗

arXiv2016

C*-simplicity of locally compact Powers groups

In this article we initiate research on locally compact C*-simple groups. We first show that every C*-simple group must be totally disconnected. Then we study C*-algebras and von Neumann algebras associated with certain groups acting on trees. After formulating a locally compact analogue of Powers' property, we prove that the reduced group C*-algebra of such groups is simple. This is the first simplicity result for C*-algebras of non-discrete groups and answers a question of de la Harpe. We also consider group von Neumann algebras of certain non-discrete groups acting on trees. We prove factoriality, determine their type and show non-amenability. We end the article by giving natural examples of groups satisfying the hypotheses of our work.

math.OA↗

arXiv2016

Cocompact amenable closed subgroups: weakly inequivalent representations in the left-regular representation

We show that if $H \leq G$ is a closed amenable and cocompact subgroup of a unimodular locally compact group, then the reduced group C*-algebra of $G$ is not simple. Equivalently, there are unitary representations of $G$ that are weakly contained in the left-regular representation, but not weakly equivalent to it. We discuss applications of this result and pose the problem to construct non-discrete topologically simple groups with a cocompact amenable closed subgroup but without a Gelfand pair.

math.GR↗

arXiv2015

Asymptotic structure of free Araki-Woods factors

The purpose of this paper is to investigate the structure of Shlyakhtenko's free Araki-Woods factors using the framework of ultraproduct von Neumann algebras. We first prove that all the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ are $ω$-solid in the following sense: for every von Neumann subalgebra $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that is the range of a faithful normal conditional expectation and such that the relative commutant $Q' \cap M^ω$ is diffuse, we have that $Q$ is amenable. Next, we prove that the continuous cores of the free Araki-Woods factors $Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ associated with mixing orthogonal representations $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ are $ω$-solid type ${\rm II_\infty}$ factors. Finally, when the orthogonal representation $U : \mathbb R \to \mathcal O(H_{\mathbb R})$ is weakly mixing, we prove a dichotomy result for all the von Neumann subalgebras $Q \subset Γ(H_{\mathbb R}, U_t)^{\prime \prime}$ that are globally invariant under the modular automorphism group $(σ_t^{φ_U})$ of the free quasi-free state $φ_U$.

math.OA↗

arXiv2015

Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity

We introduce an algebraic invariant for aperiodic inclusions of probability measure preserving equivalence relations. We use this invariant to prove that every stable orbit equivalence between free pmp actions of direct products of non-amenable Baumslag-Solitar groups whose canonical subgroup acts aperiodically forces the number of factors of the products to be the same and the factors to be isomorphic after permutation. This generalises some of the results obtained by Kida and moreover provides new measure equivalence rigidity phenomena for Baumslag-Solitar groups. We also obtain a complete classification of direct products of relative profinite completions of Baumslag-Solitar groups, continuing recent work of Elder and Willis.

math.GR↗

arXiv2014

Amalgamated free product type III factors with at most one Cartan subalgebra

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1 \ast_B M_2$ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $(M_1, φ_1) \ast (M_2, φ_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. Next, we prove that any countable nonsingular ergodic equivalence relation $\mathcal R$ defined on a standard measure space and which splits as the free product $\mathcal R = \mathcal R_1 \ast \mathcal R_2$ of recurrent subequivalence relations gives rise to a nonamenable factor $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes Γ$ arising from nonsingular free ergodic actions $Γ\curvearrowright (X, μ)$ on standard measure spaces of amalgamated groups $Γ= Γ_1 \ast_Σ Γ_2$ over a finite subgroup $Σ$.

math.OA↗

arXiv2014

A Connection between Easy Quantum Groups, Varieties of Groups and Reflection Groups

We present a link between easy quantum groups, discrete groups and combinatorics. By this, we infer new connections between quantum isometry groups, reflection groups, varieties of groups and the combinatorics of partitions. More precisely, we consider easy quantum groups and find a relation to subgroups of the infinite free product $\mathbb Z_2^{*\infty}$ of $\mathbb Z_2=\mathbb Z/2\mathbb Z$. We obtain a link with reflection groups and thus with varieties of groups, which yields a statement on the complexity of the class of easy quantum groups on the one hand, and a "quantum invariant" for varieties of groups on the other hand. Moreover, we reveal a triangular relationship between easy quantum groups, categories of partitions and discrete groups (reflection groups). As a by-product, we obtain a large number of new quantum isometry groups.

math.QA↗

arXiv2014

Easy quantum groups and quantum subgroups of a semi-direct product quantum group

We consider compact matrix quantum groups whose fundamental corepresentation matrix has entries which are partial isometries with central support. We show that such quantum groups have a simple representation as semi-direct product quantum groups of a group dual quantum group by an action of a permutation group. This general result allows us to completely classify easy quantum groups with the above property by certain reflection groups. We give four applications of our result. First, there are uncountably many easy quantum groups. Second, there are non-easy quantum groups between the free orthogonal quantum group and the permutation group. Third, we study operator algebraic properties of the hyperoctahedral series. Finally, we prove a generalised de Finetti theorem for easy quantum groups in the scope of this article.

math.QA↗

arXiv2013

The full classification of orthogonal easy quantum groups

In 1987, Woronowicz gave a definition of compact matrix quantum groups generalizing compact Lie groups in the setting of noncommutative geometry. About twenty years later, Banica and Speicher isolated a class of compact matrix quantum groups with an intrinsic combinatorial structure. These so called easy quantum groups are determined by categories of partitions. They have been proven useful in order to understand various aspects of quantum groups, in particular linked with Voiculescu's free probability theory. Furthermore, they exhibit a way to find examples of compact quantum groups besides q-deformations and quantum isometry groups. These characteristics naturally motivated attempts to fully classify them. This is completed in the present article.

math.QA↗

arXiv2013

The combinatorics of an algebraic class of easy quantum groups

Easy quantum groups are compact matrix quantum groups, whose intertwiner spaces are given by the combinatorics of categories of partitions. This class contains the symmetric group and the orthogonal group as well as Wang's quantum permutation group and his free orthogonal quantum group. In this article, we study a particular class of categories of partitions to each of which we assign a subgroup of the infinite free product of the cyclic group of order two. This is an important step in the classification of all easy quantum groups and we deduce that there are uncountably many of them. We focus on the combinatorial aspects of this assignment, complementing the quantum algebraic point of view presented in another article.

math.QA↗

arXiv2013

Tensor C*-categories arising as bimodule categories of II_1 factors

We prove that if C is a tensor C*-category in a certain class, then there exists an uncountable family of pairwise non stably isomorphic II_1 factors (M_i) such that the bimodule category of M_i is equivalent to C for all i. In particular, we prove that every finite tensor C*-category is the bimodule category of a II_1 factor. As an application we prove the existence of a II_1 factor for which the set of indices of finite index irreducible subfactors is {1, \frac{5 + \sqrt{13}}{2}, 12 + 3\sqrt{13}, 4 + \sqrt{13}, \frac{11 + 3\sqrt{13}}{2}, \frac{13 + 3\sqrt{13}}{2}, \frac{19 + 5\sqrt{13}}{2}, \frac{7 + \sqrt{13}}{2}}. We also give the first example of a II_1 factor M such that Bimod(M) is explicitly calculated and has an uncountable number of isomorphism classes of irreducible objects.

math.OA↗

arXiv2012

On the classification of free Bogoljubov crossed product von Neumann algebras by the integers

We consider crossed product von Neumann algebras arising from free Bogoljubov actions of the integers. We describe several presentations of them as amalgamated free products and cocycle crossed products and give a criterion for factoriality. A number of isomorphism results for free Bogoljubov crossed products are proved, focusing on those arising from almost periodic representations. We complement our isomorphism results by rigidity results yielding non-isomorphic free Bogoljubov crossed products and by a partial characterisation of strong solidity of a free Bogoljubov crossed products in terms of properties of the orthogonal representation from which it is constructed

math.OA↗

arXiv2012

Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions

We give an elementary proof for Lewis Bowen's theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K and every free product Γof n infinite amenable groups, the factor K^Γ/K of the Bernoulli action of Γon K^Γ by the diagonal action of K, is isomorphic with a Bernoulli action of Γ.

math.OA↗

arXiv2011

Isomorphisms and Fusion Rules of Orthogonal Free Quantum Groups and their Complexifications

We show that all orthogonal free quantum groups are isomorphic to variants of the free orthogonal Wang algebra, the hyperoctahedral quantum group or the quantum permutation group. We also obtain a description of their free complexification. In particular we complete the calculation of fusion rules of all orthogonal free quantum groups and their free complexifications.

math.OA↗