Hyperreflexivity of the derivation space of some group algebras
type: Item
instance of: scholarly article
title: Hyperreflexivity of the derivation space of some group algebras
zbMATH Open document ID: 1203.47019
author: J. Extremera
author: Jerónimo Alaminos Prats
author: Armando R. Villena
published in: Mathematische Zeitschrift
publication date: 2010-10-08T00:00:00Z
review text: Let \(A\) be a complex Banach algebra and \(X\) be a Banach \(A\)-bimodule. A linear map \(D:A\to X\) is called a derivation if \(D(ab)=D(a)\cdot b+a\cdot D(b)\) holds for arbitrary \(a, b\in A\). For instance, let \(x\in X\). Then \(\text{ad}_{x}(a)=a\cdot x - x\cdot a\) \((a\in A)\) defines a continuous derivation \(\text{ad}_{x}\); derivations of this type are called inner derivations. Denote by \(\text{Der}(A,X)\) the space of all continuous derivations from \(A\) to \(X\); in particular \(\text{Der}(A)\) is the space of all continuous derivations on \(A\). A Banach algebra \(A\) is said to be amenable if, for every dual \(A\)-bimodule \(X^*\), each element of \(\text{Der}(A,X^*)\) is an inner derivation. For a bounded linear operator \(T:A\to X\), let \(\text{der}(T)=\sup\{ \| T(ab)-T(a)\cdot b-a\cdot T(b)\|:a,b \in A, \| a\|=\| b\|=1\}\). If \(A\) is an amenable Banach algebra, then there exists a constant \(C>0\) such that, for every dual Banach \(A\)-bimodule \(X^*\) and for every bounded linear operator \(T:A\to X^*\), there exists \(\varphi \in X^*\) such that \(\| T- \text{ad}_{\varphi}\| \leq C \operatorname{der}(T)\). It follows from this that, under some additional conditions on \(A\), which are satisfied by a group algebra of an amenable group and by the algebra of compact operators on a separable Hilbert space, for instance, one has \(\text{dist}(T,\text{Der}(A))\leq C\operatorname{der}(T)\) for every bounded linear operator \(T\) on \(A\). Let \(X\) and \(Y\) be Banach spaces and let \(B(X,Y)\) be the Banach space of all bounded linear operators from \(X\) to \(Y\). The Arveson distance of \(T\in B(X,Y)\) to a closed linear subspace \({\mathcal S}\subseteq B(X,Y)\) is defined as \(\alpha(T,{\mathcal S})=\sup_{\| x\| \leq 1}\inf_{S\in {\mathcal S}}\| Tx-Sx\|\). One has \(\alpha(T,{\mathcal S}) \leq \text{dist}(T,{\mathcal S})\) for every space \({\mathcal S}\subseteq B(X,Y)\) and every \(T\in B(X,Y)\). If there exists a constant \(C\geq 1\) such that…
reviewed by: Janko Bračič
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 43A20
Mathematics Subject Classification ID: 46H05
zbMATH DE Number: 5797336
zbMATH Keywords: amenable Banach algebra
zbMATH Keywords: amenable group
zbMATH Keywords: group algebra
zbMATH Keywords: derivation
zbMATH Keywords: approximate derivation
zbMATH Keywords: approximate derivation on zero products
zbMATH Keywords: hyperreflexive linear space
MaRDI profile type: Publication
full work available at URL: s00209-009-0586-8
OpenAlex ID: W2006544135
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DOI: 10.1007/S00209-009-0586-8
description: scientific article; zbMATH DE number 5797336