On orientation and dynamics in operator algebras. I
type: Item
instance of: scholarly article
title: On orientation and dynamics in operator algebras. I
zbMATH Open document ID: 0918.46060
author: Erik M. Alfsen
author: Frederic W. Shultz
published in: Communications in Mathematical Physics
publication date: 1999-08-04T00:00:00Z
review text: Let \(A\) be a unital JB-algebra. A \(C^*\)-product (\(W^*\)-product) compatible with \(A\) is an associative product \(xy\) on the complex linear space \(A+iA\) which induces the given Jordan product on \(A\) and organizes \(A+iA\) to a \(C^*\)-algebra (von Neumann algebra) with the involution \((a+ ib)^*= a- ib\) and the norm \(\| x\|=\| x^* x\|^{1/2}\). It is clear that a JB-algebra \(A\) is the selfadjoint part of a \(C^*\)-algebra if and only if there exists a \(C^*\)-product compatible with \(A\) on \(A+ iA\), similarly in the JBW-context. The main theorem of the paper proves: A unital JB-algebra \(A\) is isomorphic to the selfadjoint part of a \(C^*\)-algebra if and only if there exists a dynamical correspondence on \(A\). Moreover, each dynamical correspondence gives the ``Lie part'' of a unique \(C^*\)-product compatible with \(A\), and each \(C^*\)-product compatible with \(A\) appears in this way. The same conclusions hold with ``JBW'' in place of ``JB'' and ``\(W^*\)'' or ``von Neumann'' in place of ``\(C^*\)''. The paper also explains how the associative product is determined by a general notion of orientation which is related to dynamics. This concept of orientation bridges an approach in Connes' characterization of the natural cone of a von Neumann algebra and an authors' characterization of the state space of a \(C^*\)-algebra.
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1214083
zbMATH Keywords: dynamical correspondence
zbMATH Keywords: Lie part
zbMATH Keywords: \(W^*\)-product
zbMATH Keywords: unital JB-algebra
zbMATH Keywords: \(C^*\)-product
zbMATH Keywords: Jordan product
zbMATH Keywords: orientation
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: state space
reviewed by: Antonio Fernández López
MaRDI profile type: Publication
full work available at URL: s002200050350
OpenAlex ID: W2071617550
DOI: 10.1007/S002200050350
description: scientific article; zbMATH DE number 1214083