Inner \(M\)-ideals in Banach algebras
type: Item
instance of: scholarly article
title: Inner \(M\)-ideals in Banach algebras
zbMATH Open document ID: 0726.46030
DOI: 10.1007/BF01445200
author: Wend Werner
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164859
review text: We characterize Banach spaces \(X\) satisfying the relation \[ L(X)^*=(X^*{\hat \otimes}_{\pi}X^{**})\oplus_ 1(X{\hat \otimes}_{\epsilon}X^*)^{\perp} \] in terms of the asymptotic behaviour of certain nets of finite dimensional operators converging strongly to the identity on \(X\). Central for our considerations is the concept of an inner \(M\)-ideal, which to some extend is investigated systematically. The results thus obtained allow also to describe the \(M\)-ideals in some algebras of bounded operators explicitly. For example, it is shown that \(J\) is an \(M\)-ideal of \(L({\mathfrak A})\), the space of bounded operators on a function algebra \({\mathfrak A}\), if and only if \[ J=\{T\in L({\mathfrak A})| \quad \limsup_{k\to D}\| T^*\delta_ k\| =0\}, \] where \(D\) runs through the \(p\)-sets of \({\mathfrak A}\). An important tool in our reasoning is a - seemingly unknown - result stating that any operator on \({\mathfrak A}\) with an `almost real' numerical range is the perturbation of a multiplication operator.
reviewed by: Wend Werner
Mathematics Subject Classification ID: 46H10
Mathematics Subject Classification ID: 46H20
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46M05
zbMATH DE Number: 4198820
zbMATH Keywords: almost real numerical range
zbMATH Keywords: inner M-ideal
zbMATH Keywords: M-ideals in some algebras of bounded operators
zbMATH Keywords: perturbation of a multiplication operator
MaRDI profile type: Publication
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OpenAlex ID: W2313515041