Automorphisms of central extensions of type I von Neumann algebras

Details Automorphisms of central extensions of type I von Neumann algebras Automorphisms of central extensions of type I von Neumann algebras

2011-04-25

arxiv.org/abs/1104.4698v1

Mathematics

Operator Algebras

16 pages

Scientific paper

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.

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