Skew commutators and Lie isomorphisms in real von Neumann algebras
type: Item
instance of: scholarly article
title: Skew commutators and Lie isomorphisms in real von Neumann algebras
zbMATH Open document ID: 0903.46057
published in: Journal of Functional Analysis
publication date: 1999-01-07T00:00:00Z
review text: Let \(R\) be a real von Neumann algebra. Then its selfadjoint elements \(R_s= \{x\in R, x^*= x\}\) form a weakly closed Jordan algebra, i.e., JW-algebra with respect to symmetrized product \(x\circ y= xy+ yx\), while its skew-adjoint elements \(R_k= \{x\in R, x^*= -x\}\) form a real Lie algebra with brackets \([x,y]= xy- yx\). It was proved by the author in an earlier paper that if \(R\) and \(S\) are real factors nonisomorphic to \(\mathbb{R}\), then the JW-algebras \(R_s\) and \(S_s\) are isomorphic if and only if \(R\) and \(S\) are (associative) \(*\)-isomorphic. In the present paper, the author considers a similar problem for Lie algebras \(R_k\) and \(S_k\). Denote \([R_k, R_k]\) the derived Lie ring, i.e., the additive span of all commutators \([x,y]\), where \(x\), \(y\in R_k\); it is a Lie ideal in the Lie ring \(R_k\). The main result states that if \(R\) and \(S\) are real factors not of type \(I_1\) and \(I_2\) then \([R_k, R_k]\) and \([S_k, S_k]\) are isomorphic if and only if \(R\) and \(S\) are \(*\)-isomorphic. In particular, \(R_k\) and \(S_k\) are Lie isomorphic if and only if \(R\) and \(S\) are \(*\)-isomorphic. Also, an application to Lie algebras of form \(M^\alpha(- 1)= \{x\in M, \alpha(x)= -x\}\) is given, where \(\alpha\) is a \(*\)-anti-automorphism of order 2 of a (complex) von Neumann algebra \(M\).
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L70
zbMATH DE Number: 889772
zbMATH Keywords: commutator
zbMATH Keywords: \(*\)-isomorphism
zbMATH Keywords: weakly closed Jordan algebra
zbMATH Keywords: JW-algebra
zbMATH Keywords: symmetrized product
zbMATH Keywords: Lie algebras
zbMATH Keywords: \(*\)-anti-automorphism
zbMATH Keywords: von Neumann algebra
author: Sh. A. Ayupov
reviewed by: Jinchuan Hou
MaRDI profile type: Publication
full work available at URL: jfan.1996.0060
OpenAlex ID: W1979324158
DOI: 10.1006/JFAN.1996.0060
description: scientific article; zbMATH DE number 889772