On Defining Aw∗-Algebras and Rickart C∗-Algebras

saitô, kazuyuki; wright, j. d. maitland;

Let A be a C∗-algebra. It is shown that A is an AW∗-algebra if, and only if, each maximal abelian self-adjoint (m.a.s.a.) 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
m.a.s.a. subalgebra of A is monotone σ-complete.