On Defining AW*-algebras and Rickart C*-algebras
Kazuyuki Sait\^o, J.D. Maitland Wright

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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Algebraic structures and combinatorial models
