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Automorphisms of central extensions of type I von Neumann algebras

By S. Albeverio, Sh. A. Ayupov, K. K. Kudaybergenov and R. T. Djumamuratov

Abstract

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ For type I von Neumann algebras $E(M)$ coincides with the algebra $LS(M)$ of all locally measurable operators affiliated with $M.$ In this case we show that an arbitrary automorphism $T$ of $E(M)$ can be decomposed as $T=T_a\circ T_\phi,$ where $T_a(x)=axa^{-1}$ is an inner automorphism implemented by an element $a\in E(M),$ and $T_\phi$ is a special automorphism generated by an automorphism $\phi$ of the center of $E(M).$ In particular if $M$ is of type I$_\infty$ then every band preserving automorphism of $E(M)$ is inner.Comment: 16 page

Topics: Mathematics - Operator Algebras, Primary 46L40, Secondary 46L51, 46L57
Year: 2011
OAI identifier: oai:arXiv.org:1104.4698
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