Real \(W^ *\)-algebras with an abelian Hermitian part.
type: Item
instance of: scholarly article
title: Real \(W^ *\)-algebras with an abelian Hermitian part.
zbMATH Open document ID: 1036.46045
DOI: 10.1023/A:1014863227576
author: A. Kh. Abduvaitov
author: Sh. A. Ayupov
author: A. A. Rakhimov
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: The present paper is devoted to the description of real \(W^{*}\)-algebras with abelian hermitian part. A real \(W^{*}\)-algebra is a real \(C^{*}\)-algebra \(\mathcal{R}\) such that \(\mathcal{R}+i\mathcal{R}\) is a complex \(W^{*}\)-algebra. The authors show that any real \(W^{*}\)-algebra with abelian hermitian part can be decomposed into the direct sum of two real \(W^{*}\)-algebras one of which is abelian and the other has a complex enveloping algebra of type \(I_2\). Then, showing that any real \(W^{*}\)-algebra with abelian hermitian part and a complex enveloping algebra of type \(I_2\) is \(*\)-isomorphic to \(L_{\mathbb{Q}}^\infty (\Omega ,\nu )\) (where \(\Omega \) is a hyper-Stonean compactum with Radon measure \(\nu \) and \(\mathbb{Q}\) is the skew field of quaternions) and using the fact that an abelian real \(W^{*}\)-algebra is \(*\)-isomorphic to a direct sum of algebras of the form \(L_{\mathbb{R}}^\infty (\Omega ,\nu )\) and \(L_{\mathbb{C}}^\infty (\Omega ,\nu )\), they obtain the description of real \(W^{*}\)-algebras with abelian hermitian part.
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46K05
zbMATH DE Number: 1928186
zbMATH Keywords: real \(W^{*}\)-algebras
zbMATH Keywords: real \(C^{*}\)-algebras
zbMATH Keywords: Hermitian part of a real \(W^{*}\)-algebra
zbMATH Keywords: involutive antiautomorphisms
reviewed by: Maria Joiţa
MaRDI profile type: Publication
full work available at URL: a:1014863227576
OpenAlex ID: W264998573
description: scientific article; zbMATH DE number 1928186