arXiv subjects

Tim de Laat

Publications and source records attributed to Tim de Laat.

At least 19 recordsLinked to original sources

arXiv2024

Entropies and Poisson boundaries of random walks on groups with rapid decay

Let $G$ be a countable group and $\mu$ a probability measure on $G$. We build a new framework to compute asymptotic quantities associated with the $\mu$-random walk on $G$, using methods from harmonic analysis on groups and Banach space theory, most notably complex interpolation. It is shown that under mild conditions, the Lyapunov exponent of the $\mu$-random walk with respect to a weight $\omega$ on $G$ can be computed in terms of the asymptotic behavior of the spectral radius of $\mu$ in an ascending class of weighted group algebras, and we prove that for natural choices of $\omega$ and $\mu$, the Lyapunov exponent vanishes. Also, we show that the Avez entropy of the $\mu$-random walk can be realized as the Lyapunov exponent of $\mu$ with respect to a suitable weight. We apply our results to stationary dynamical systems consisting of an action of a group with the property of rapid decay on a probability space. We prove that whenever the associated Koopman representation is weakly contained in the left-regular representation of the group, then the Avez entropy coincides with the Furstenberg entropy of the stationary space. This gives a characterization of (Zimmer) amenability for actions of rapid decay groups on stationary spaces. Next, by considering the spectral radius in the algebras of $p$-pseudofunctions on $G$, we introduce a new asymptotic quantity, which we call convolution entropy. We show that for groups with the property of rapid decay, the convolution entropy coincides with the Avez entropy.

math.DS↗

arXiv2024

Spectral gap and origami expanders

We construct the first measure-preserving affine actions with spectral gap on surfaces of arbitrary genus $g > 1$. We achieve this by finding geometric representatives of multi-twists on origami surfaces. As a major application, we construct new expanders that are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. Our methods also show that the Margulis expander, and hence the Gabber--Galil expander, is coarsely distinct from the Selberg expander.

math.MG↗

arXiv2023

Dynamical propagation and Roe algebras of warped spaces

Given a non-singular action $\Gamma \curvearrowright (X,\mu)$, we define the $*$-algebra $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ of operators of finite dynamical propagation associated with this action. This assignment is completely canonical and only depends on the measure class of $\mu$. We prove that the algebraic crossed product $L^{\infty}X \rtimes_{\rm alg} \Gamma$ surjects onto $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and that this surjection is a $\ast$-isomorphism whenever the action is essentially free. As a consequence, we canonically characterize ergodicity and strong ergodicity of the action in terms of structural properties of $\mathbb C_{\rm fp}[\Gamma \curvearrowright X]$ and its closure. We also use these techniques to describe the Roe algebra of a warped space in terms of the Roe algebra of the (non-warped) space and the group action. We apply this result to Roe algebras of warped cones.

math.OA↗

arXiv2023

Unitary $L^{p+}$-representations of almost automorphism groups

Let $G$ be a locally compact group with an open subgroup $H$ with the Kunze-Stein property, and let $π$ be a unitary representation of $H$. We show that the representation $\widetildeπ$ of $G$ induced from $π$ is an $L^{p+}$-representation if and only if $π$ is an $L^{p+}$-representation. We deduce the following consequence for a large natural class of almost automorphism groups $G$ of trees: For every $p \in (2,\infty)$, the group $G$ has a unitary $L^{p+}$-representation that is not an $L^{q+}$-representation for any $q < p$. This in particular applies to the Neretin groups.

math.RT↗

arXiv2023

Actions of higher rank groups on uniformly convex Banach spaces

We prove that all isometric actions of higher rank simple Lie groups and their lattices on arbitrary uniformly convex Banach spaces have a fixed point. This vastly generalises a recent breakthrough of Oppenheim. Combined with earlier work of Lafforgue and of Liao on strong Banach property (T) for non-Archimedean higher rank simple groups, this confirms a long-standing conjecture of Bader, Furman, Gelander and Monod. As a consequence, we deduce that sequences of Cayley graphs of finite quotients of a higher rank lattice are super-expanders.

math.GR↗

arXiv2021

Weak*-continuity of invariant means on spaces of matrix coefficients

With every locally compact group $G$, one can associate several interesting bi-invariant subspaces $X(G)$ of the weakly almost periodic functions $\mathrm{WAP}(G)$ on $G$, each of which captures parts of the representation theory of $G$. Under certain natural assumptions, such a space $X(G)$ carries a unique invariant mean and has a natural predual, and we view the weak$^*$-continuity of this mean as a rigidity property of $G$. Important examples of such spaces $X(G)$, which we study explicitly, are the algebra $M_{\mathrm{cb}}A_p(G)$ of $p$-completely bounded multipliers of the Figà-Talamanca-Herz algebra $A_p(G)$ and the $p$-Fourier-Stieltjes algebra $B_p(G)$. In the setting of connected Lie groups $G$, we relate the weak$^*$-continuity of the mean on these spaces to structural properties of $G$. Our results generalise results of Bekka, Kaniuth, Lau and Schlichting.

math.FA↗

arXiv2020

Group $C^*$-algebras of locally compact groups acting on trees

We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from $L^p$-integrability properties of matrix coefficients of unitary representations - of locally compact groups $G$ acting on (semi-)homogeneous trees of sufficiently large degree. These group $C^*$-algebras lie between the universal and the reduced group $C^*$-algebra. By directly investigating these $L^p$-integrability properties, we first show that for every non-compact, closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a (semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$ and every $2 \leq q < p \leq \infty$, the canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ is not injective. This reproves a result of Samei and Wiersma. We prove that under the additional assumptions that $G$ acts transitively on $T$ and that it has Tits' independence property, the group $C^*$-algebras $C^*_{L^{p+}}(G)$ are the only group $C^*$-algebras coming from $G$-invariant ideals in the Fourier-Stieltjes algebra $B(G)$. Additionally, we show that given a group $G$ as before, every group $C^*$-algebra $C^*_μ(G)$ that is distinguishable (as a group $C^*$-algebra) from the universal group $C^*$-algebra of $G$ and whose dual space $C^*_μ(G)^*$ is a $G$-invariant ideal in $B(G)$ is abstractly ${}^*$-isomorphic to the reduced group $C^*$-algebra of $G$.

math.OA↗

arXiv2020

Exotic group $C^*$-algebras of simple Lie groups with real rank one

Exotic group $C^*$-algebras are $C^*$-algebras that lie between the universal and the reduced group $C^*$-algebra of a locally compact group. We consider simple Lie groups $G$ with real rank one and investigate their exotic group $C^{*}$-algebras $C^*_{L^{p+}}(G)$, which are defined through $L^p$-integrability properties of matrix coefficients of unitary representations. First, we show that the subset of equivalence classes of irreducible unitary $L^{p+}$-representations forms a closed ideal of the unitary dual of these groups. This result holds more generally for groups with the Kunze-Stein property. Second, for every classical simple Lie group $G$ with real rank one and every $2 \leq q < p \leq \infty$, we determine whether the canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ has non-trivial kernel. Our results generalize, with different methods, recent results of Samei and Wiersma on exotic group $C^*$-algebras of $\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$. In particular, our approach also works for groups with property (T).

math.OA↗

arXiv2019

Banach space actions and $L^2$-spectral gap

Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap $> \frac{1}{2}$, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example an $L^p$-space with $1 < p < \infty$ or an interpolation space between a Hilbert space and an arbitrary Banach space) as soon as the Laplacian on the links has a two-sided spectral gap $>1-\varepsilon$. This criterion applies to random groups in the triangular density model for densities $> \frac{1}{3}$. In this way, we are able to generalize recent results of Druţu and Mackay to affine isometric actions of random groups on uniformly curved Banach spaces. Also, in the setting of actions on $L^p$-spaces, our results are quantitatively stronger, even in the case $p=2$. This naturally leads to new estimates on the conformal dimension of the boundary of random groups in the triangular model. Additionally, we obtain results on the eigenvalues of the $p$-Laplacian on graphs, and on the spectrum and degree distribution of Erdős-Rényi graphs.

math.FA↗

arXiv2018

Superexpanders from group actions on compact manifolds

It is known that the expanders arising as increasing sequences of level sets of warped cones, as introduced by the second-named author, do not coarsely embed into a Banach space as soon as the corresponding warped cone does not coarsely embed into this Banach space. Combining this with non-embeddability results for warped cones by Nowak and Sawicki, which relate the non-embeddability of a warped cone to a spectral gap property of the underlying action, we provide new examples of expanders that do not coarsely embed into any Banach space with nontrivial type. Moreover, we prove that these expanders are not coarsely equivalent to a Lafforgue expander. In particular, we provide infinitely many coarsely distinct superexpanders that are not Lafforgue expanders. In addition, we prove a quasi-isometric rigidity result for warped cones.

math.MG↗

arXiv2018

The Fourier algebra of a rigid $C^{\ast}$-tensor category

Completely positive and completely bounded mutlipliers on rigid $C^{\ast}$-tensor categories were introduced by Popa and Vaes. Using these notions, we define and study the Fourier-Stieltjes algebra, the Fourier algebra and the algebra of completely bounded multipliers of a rigid $C^{\ast}$-tensor category. The rich structure that these algebras have in the setting of locally compact groups is still present in the setting of rigid $C^{\ast}$-tensor categories. We also prove that Leptin's characterization of amenability still holds in this setting, and we collect some natural observations on property (T).

math.OA↗

arXiv2017

Howe-Moore type theorems for quantum groups and rigid C*-tensor categories

We formulate and study Howe-Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast}$-tensor categories. We say that a rigid $C^{\ast}$-tensor category $\mathcal{C}$ has the Howe-Moore property if every completely positive multiplier on $\mathcal{C}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe-Moore property. As an immediate consequence, we deduce the Howe-Moore property for Temperley-Lieb-Jones standard invariants with principal graph $A_{\infty}$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan-Wenzl categories. Additionally, in the specific case of the quantum groups $\mathrm{SU}_q(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\mathrm{SU}_q(N)$, which coincide with the completely positive multipliers on the representation category of $\mathrm{SU}_q(N)$.

math.OA↗

arXiv2016

A complete characterization of connected Lie groups with the Approximation Property

We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.

math.GR↗

arXiv2016

On strong property (T) and fixed point properties for Lie groups

We consider certain strengthenings of property (T) relative to Banach spaces that are satisfied by high rank Lie groups. Let X be a Banach space for which, for all k, the Banach--Mazur distance to a Hilbert space of all k-dimensional subspaces is bounded above by a power of k strictly less than one half. We prove that every connected simple Lie group of sufficiently large real rank depending on X has strong property (T) of Lafforgue with respect to X. As a consequence, we obtain that every continuous affine isometric action of such a high rank group (or a lattice in such a group) on X has a fixed point. This result corroborates a conjecture of Bader, Furman, Gelander and Monod. For the special linear Lie groups, we also present a more direct approach to fixed point properties, or, more precisely, to the boundedness of quasi-cocycles. Without appealing to strong property (T), we prove that given a Banach space X as above, every special linear group of sufficiently large rank satisfies the following property: every quasi-1-cocycle with values in an isometric representation on X is bounded.

math.FA↗

arXiv2015

Approximation properties for noncommutative $L^p$-spaces of high rank lattices and nonembeddability of expanders

This article contains two rigidity type results for $\mathrm{SL}(n,\mathbb{Z})$ for large $n$ that share the same proof. Firstly, we prove that for every $p \in [1,\infty]$ different from $2$, the noncommutative $L^p$-space associated with $\mathrm{SL}(n,\mathbb{Z})$ does not have the completely bounded approximation property for sufficiently large $n$ depending on $p$. The second result concerns the coarse embeddability of expander families constructed from $\mathrm{SL}(n,\mathbb{Z})$. Let $X$ be a Banach space and suppose that there exist $β< \frac{1}{2}$ and $C>0$ such that the Banach-Mazur distance to a Hilbert space of all $k$-dimensional subspaces of $X$ is bounded above by $C k^β$. Then the expander family constructed from $\mathrm{SL}(n,\mathbb{Z})$ does not coarsely embed into $X$ for sufficiently large $n$ depending on $X$. More generally, we prove that both results hold for lattices in connected simple real Lie groups with sufficiently high real rank.

math.OA↗

arXiv2015

Simple Lie groups without the Approximation Property II

We prove that the universal covering group $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$ of $\mathrm{Sp}(2,\mathbb{R})$ does not have the Approximation Property (AP). Together with the fact that $\mathrm{SL}(3,\mathbb{R})$ does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that $\mathrm{Sp}(2,\mathbb{R})$ does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptation of the methods we use to study the AP, we obtain results on approximation properties for noncommutative $L^p$-spaces associated with lattices in $\widetilde{\mathrm{Sp}}(2,\mathbb{R})$. Combining this with earlier results of Lafforgue and de la Salle and results of the second named author of this article, this gives rise to results on approximation properties of noncommutative $L^p$-spaces associated with lattices in any connected simple Lie group.

math.OA↗

arXiv2015

Strong property (T) for higher rank simple Lie groups

We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces $\mathcal{E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that $\mathrm{SL}(3,\mathbb{R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank groups, it is sufficient to prove strong property (T) for $\mathrm{Sp}(2,\mathbb{R})$ and its universal covering group. As consequences of our main result, it follows that for $X \in \mathcal{E}_{10}$, connected higher rank simple Lie groups and their lattices have property (F$_X$) of Bader, Furman, Gelander and Monod, and that the expanders contructed from a lattice in a connected higher rank simple Lie group do not admit a coarse embedding into $X$.

math.GR↗

arXiv2013

Approximation properties for noncommutative Lp-spaces associated with lattices in Lie groups

In 2010, Lafforgue and de la Salle gave examples of noncommutative Lp-spaces without the operator space approximation property (OAP) and, hence, without the completely bounded approximation property (CBAP). To this purpose, they introduced the property of completely bounded approximation by Schur multipliers on Sp and proved that for p < 4/3 and p > 4 the groups SL(n,Z), with n \geq 3, do not have it. Since for 1 < p < \infty the property of completely bounded approximation by Schur multipliers on Sp is weaker than the approximation property of Haagerup and Kraus (AP), these groups were also the first examples of exact groups without the AP. Recently, Haagerup and the author proved that also the group Sp(2,R) does not have the AP, without using the property of completely bounded approximation by Schur multipliers on Sp. In this paper, we prove that Sp(2,R) does not have the property of completely bounded approximation by Schur multipliers on Sp for p < 12/11 and p > 12. It follows that a large class of noncommutative Lp-spaces does not have the OAP or CBAP.

math.OA↗