A classification of type I \(AW^*\)-algebras and Boolean valued analysis (Q1080130)

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.