On a Local Structure in Kaplansky Algebras. Definitions and Basic Properties

TL;DR

This paper introduces locally AW*-algebras as a generalization of Kaplansky's AW*-algebras, establishing their basic properties, spectral theorem, and connections to Arens-Michael decompositions.

Contribution

It defines locally AW*-algebras, explores their properties, and proves a spectral theorem, extending the theory of AW*-algebras to a local setting.

Findings

A locally C*-algebra is a locally AW*-algebra iff its Arens-Michael decomposition consists of AW*-algebras.

The bounded part of a locally AW*-algebra is an AW*-algebra.

A spectral theorem holds for locally AW*-algebras.

Abstract

We introduce and study locally AW*-algebras (Baer locally C*-algebras) as a locally multiplicatively-convex generalization of AW*-algebras of Kaplansky. Among other basic properties of these algebras, it is established that: {\bullet} A locally C*-algebra is a locally AW*-algebra iff there exists its Arens-Michael decomposition consisting entirely of AW*-algebras; {\bullet} A bounded part of a locally AW*-algebra is an AW*-algebra; {\bullet} The Spectral Theorem for locally AW*-algebras.