Derivations on the algebra of operators in Hilbert \(C^*\)-modules
type: Item
instance of: scholarly article
title: Derivations on the algebra of operators in Hilbert \(C^*\)-modules
zbMATH Open document ID: 1263.47042
author: Pengtong Li
author: Deguang Han
author: Wai-Shing Tang
publication date: 2012-11-07T00:00:00Z
review text: A derivation of an algebra \(A\) is a linear mapping \(\delta: A \to A\) such that \(\delta(ab)=\delta(a)b+a\delta(b)\). A derivation of the form \(\delta(a)=ac_0-bc_0\) for some fixed \(c_0\in A\) is called an inner derivation. One of the interesting problems in the theory of derivations is to identify those algebras on which all the derivations are inner. It is known that every derivation of a \(W^*\)-algebra is inner; see \textit{S. Sakai} [Ann. Math. (2) 83, 273--279 (1966; Zbl 0139.30601)]. In the paper under review, the authors treat this problem in the setting of Hilbert \(C^*\)-modules. Let \(M\) be a full Hilbert \(C^*\)-module over a \(C^*\)-algebra \(A\), and let \(\mathrm{End}_A^*(M)\) be the algebra of all adjointable operators on \(M\). The authors prove that if \(A\) is unital and commutative, then every derivation of \(\mathrm{End}_A^*(M)\) is an inner derivation, and that if \(A\) is \(\sigma\)-unital and commutative, then innerness of derivations on compact operators completely decides innerness of derivations on \(\mathrm{End}_A^*(M)\). If \(A\) is unital (no commutativity is assumed) such that every derivation of \(A\) is inner, then they show that every derivation of \(\mathrm{End}_A^*(L_n(A))\) is also inner, where \(L_n(A)\) denotes the direct sum of \(n\) copies of \(A\).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46L08
Mathematics Subject Classification ID: 47B49
zbMATH DE Number: 6102781
zbMATH Keywords: derivations
zbMATH Keywords: inner derivations
zbMATH Keywords: \(C^*\)-algebras
zbMATH Keywords: Hilbert \(C^*\)-modules
MaRDI profile type: Publication
full work available at URL: s10114-012-0172-6
OpenAlex ID: W2126103881
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DOI: 10.1007/S10114-012-0172-6
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published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6102781