A classification of type I \(AW^*\)-algebras and Boolean valued analysis
type: Item
instance of: scholarly article
title: A classification of type I \(AW^*\)-algebras and Boolean valued analysis
zbMATH Open document ID: 0599.46083
author: Masanao Ozawa
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: I. Kaplansky proved that every type I \(AW^*\)-algebra could be decomposed into a direct sum of homogeneous \(AW^*\)-algebras but he stated: ''One detail has resisted complete solution thus far; the uniqueness of the cardinal number attached to a homogeneous \(AW^*\)- algebra of type I.'' Kaplansky later conjectured that this question had a negative answer. In this impressive paper, the author settles this long- standing problem. Given any infinite cardinals \(\alpha\) and \(\beta\) with \(\alpha <\beta\), there is an \(AW^*\)-algebra which is \(\gamma\)- homogeneous for each cardinal \(\gamma\) such that \(\alpha\leq \gamma \leq \beta.\) The key tools used are the Boolean-valued models of set theory introduced by Scott and Solovay. These methods have also been used in operator theory, with conspicuous-success, by G. Takeuti.
Mathematics Subject Classification ID: 46L35
Mathematics Subject Classification ID: 03C90
Mathematics Subject Classification ID: 46S20
Mathematics Subject Classification ID: 03E40
zbMATH DE Number: 3967133
zbMATH Keywords: type I \(AW^*\)-algebra
zbMATH Keywords: uniqueness of the cardinal number attached to a homogeneous \(AW^*\)-algebra of type I
zbMATH Keywords: Boolean-valued models of set theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03640589
OpenAlex ID: W2170041071
DOI: 10.2969/JMSJ/03640589
description: scientific article; zbMATH DE number 3967133