Similarity problems and completely bounded maps
type: Item
instance of: scholarly article
title: Similarity problems and completely bounded maps
zbMATH Open document ID: 0843.47010
author: Gilles Pisier
published in: Lecture Notes in Mathematics
publication date: 1995-12-13T00:00:00Z
full work available at URL: t1618.htm
review text: The lecture notes by G. Pisier are centred around the following three similarity problems: (1) Is every continuous bounded representation \(\pi\) of a locally compact group \(G\) on a Hilbert space \(H\) similar to a unitary representation? (In other words, does there exist an invertible operator \(S: H\to H\) such that the representation \(S^{- 1} \pi S\) is a unitary representation?) (2) Is every bounded unital homomorphism \(u: A\to B(H)\) mapping a \(C^*\)-algebra \(A\) into the algebra \(B(H)\) of all bounded operators acting on a Hilbert space \(H\) similar to a \(*\)-representation? (3) Is every polynomially bounded operator acting on a Hilbert space similar to a contraction? The author provides a very nice exposition of various results, concepts and counterexamples related to these problems and reflecting recent development in the theory that surrounds them. The fundamental concept of the book is the notion of complete boundedness of operators. The author shows that all similarity problems mentioned above can be placed in a common context and reformulated as a question of complete boundedness of some bounded linear operators. The text is organized into seven chapters. The first chapter is devoted to a von Neumann inequality for a contraction on a Hilbert space and its extension (due to Ando) for two commuting contractions. The interesting fact that Ando's result is not valid for three commuting contractions is commentated. Moreover, Hilbert spaces are characterized as the only Banach spases satisfying von Neumann inequality. The second chapter deals with bounded representations of discrete groups. The space \(B(G)\) of all matrix coefficients of the unitary representations of \(G\) is introduced and studied. It is shown that the free group on three generators admits a non-unitarizable uniformly bounded group representation. Besides, several characterizations of amenable groups are given in terms of function spaces defined on them. The third chapter is devote…
Mathematics Subject Classification ID: 47A65
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 825912
zbMATH Keywords: similarity problems
zbMATH Keywords: similar to a \(*\)-representation
zbMATH Keywords: polynomially bounded operator
zbMATH Keywords: similar to a constraction
zbMATH Keywords: complete boundedness of operators
zbMATH Keywords: von Neumann inequality
zbMATH Keywords: bounded representations of discrete groups
zbMATH Keywords: unitary representations
zbMATH Keywords: Paulsen's theorem
zbMATH Keywords: Schur multipliers
zbMATH Keywords: Grothendieck's inequality
zbMATH Keywords: Grothendieck constant
zbMATH Keywords: Hankel-Schur multipliers
zbMATH Keywords: Herz-Schur multipliers
zbMATH Keywords: Haagerup theorem
reviewed by: Jan Hamhalter
MaRDI profile type: Publication
DOI: 10.1007/978-3-662-21537-1
OpenAlex ID: W4300256761
description: scientific article; zbMATH DE number 825912