On Defining AW*-algebras and Rickart C*-algebras

TL;DR

This paper characterizes AW*-algebras and Rickart C*-algebras through properties of their maximal abelian self-adjoint subalgebras, providing new criteria for their identification.

Contribution

It establishes necessary and sufficient conditions for AW*-algebras and Rickart C*-algebras based on monotone completeness of subalgebras.

Findings

AW*-algebra characterized by monotone completeness of maximal abelian subalgebras

Rickart C*-algebra characterized by unitality and mbda-mbda-completeness of subalgebras

Provides new algebraic criteria for classifying these C*-algebras

Abstract

Let A be a C*-algebra. It is shown that A is an AW*-algebra if, and only if, each maximal abelian self--adjoint subalgebra of A is monotone complete. An analogous result is proved for Rickart C*-algebras; a C*-algebra is a Rickart C*-algebra if, and only if, it is unital and each maximal abelian self--adjoint subalgebra of A is monotone {\sigma}-complete.