Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra
type: Item
instance of: scholarly article
title: Derivations of von Neumann algebras into the compact ideal space of a semifinite algebra
zbMATH Open document ID: 0673.46042
DOI: 10.1215/S0012-7094-88-05722-5
author: Sorin Popa
author: Florin Rădulescu
published in: Duke Mathematical Journal
publication date: 1988-01-01T00:00:00Z
review text: Let M be a semifinite von Neumann algebra, \({\mathcal J}(M)\) the closed ideal generated by the finite projection and N a weakly closed *-subalgebra of M. N is decomposed as \(N=N_ 0\oplus N_ 1\), where \(N_ 0\) is a finite type I von Neumann algebra and \(N_ 1\) a von Neumann algebra with no finite type I summands. One of the main results is that every derivation of N into \({\mathcal J}(M)\) is inner, i.e. of the form \(\delta (x)=Kx-xK\) for some \(K\in {\mathcal J}(M)\), provided that there exists a partition of the unity \(\{p_ i\}\) in the center \({\mathcal Z}(N_ 0)\) of \(N_ 0\) such that for each i it holds that \({\mathcal Z}(N_ 0)p_ i\subset {\mathcal Z}(M)p_ i\) or \({\mathcal Z}(M)p_ i\subset {\mathcal Z}(N_ 0)p_ i.\) The other main result is the following: if \(M=L^{\infty}[0,1]{\bar \oplus}{\mathcal B}(L^ 2({\mathbb{T}}))\) and \(N=1\oplus L^{\infty}({\mathbb{T}})\subset M\), there exists an operator \(T\in M\) which commutes modulo \({\mathcal J}(M)\) with all the elements of N but which is not a \({\mathcal J}(M)\)-permutation of an element commuting with M. In particular ad T is a not inner derivation from M into \({\mathcal J}(M)\).
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 4101893
zbMATH Keywords: semifinite von Neumann
zbMATH Keywords: finite type I von Neumann algebra
zbMATH Keywords: von Neumann algebra with no finite type I summands
zbMATH Keywords: inner derivation
reviewed by: Gustavo Corach
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4101893