Related papers: The $p\,$-approximation property for simple Lie gr…

For a locally compact group G, let A(G) denote its Fourier algebra, and let M_0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can…

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,…

In this paper we consider the class of connected simple Lie groups equipped with the discrete topology. We show that within this class of groups the following approximation properties are equivalent: (1) the Haagerup property; (2) weak…

Let $G$ be the symplectic group $Sp_4$ over a non Archimedean local field of any characteristic. It is proved in this paper that for $p\in[1,4/3)\cup (4,\infty]$ neither the group $G$ nor its lattices have the property of approximation by…

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…

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…

It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C*-algebra of any lattice in a non-compact simple Lie…

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…

The weak Haagerup property for locally compact groups and the weak Haagerup constant was recently introduced by the second author. The weak Haagerup property is weaker than both weak amenability introduced by Cowling and the first author…

We study the problem of determining all connected Lie groups $G$ which have the following property (hlp): every sub-Laplacian $L$ on $G$ is of holomorphic $L^p$-type for $1\leq p<\infty, p\ne 2.$ First we show that semi-simple non-compact…

A locally compact group $G$ is said to be weakly amenable if the Fourier algebra $A(G)$ admits completely bounded approximative units. Consider the family of groups $G_n=SL(2,\Bbb R)\ltimes H_n$ where $n\ge 2$, $H_n$ is the $2n+1$…

We show that finite quasisimple groups of Lie type in characteristic $p$ with an irreducible representation of prime degree $r$ over a finite field of characteristic $p$ have orders bounded above by a function of $r$, independent of $p$. We…

In this work, we study groupoids and their approximation properties, generalizing both the definitions and some known results for the group case. More precisely, we introduce weak amenability for groupoids using the definition of the…

We extend a theorem of Haagerup and Kraus in the C*-algebra context: for a locally compact group with the approximation property (AP), the reduced C*-crossed product construction preserves the strong operator approximation property (SOAP).…

Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of…

We prove that hyperbolic groups are weakly amenable. This partially extends the result of Cowling and Haagerup showing that lattices in simple Lie groups of real rank one are weakly amenable. We take a combinatorial approach in the spirit…

Recently, a complete characterization of connected Lie groups with the Approximation Property was given. The proof used of the newly introduced property (T*). We present here a short proof of the same result avoiding the use of property…

We first give simplified and corrected accounts of some results in \cite{PiRCP} on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing \cite{Turing} to give a simplified proof that any definable…

We introduce the weak Haagerup property for locally compact groups and prove several hereditary results for the class of groups with this approximation property. The class contains a priori all weakly amenable groups and groups with the…

For any 1\leq p \leq \infty different from 2, we give examples of non-commutative Lp spaces without the completely bounded approximation property. Let F be a non-archimedian local field. If p>4 or p<4/3 and r\geq 3 these examples are the…