ON DEFINING AW*-ALGEBRAS AND RICKART 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 monotonecomplete. 1. AW*-algebras In this note A will be a C*-algebra which is assumed to have a unit element (unless we state otherwise). Let ProjA be the set of all projections in A. Let Asa be the self-adjoint part of A. We recall that the positive cone A + = {zz � : z ∈ A} induces a partial ordering on A. Since each projection is in A + , it follows that the partial ordering of Asa induces a partial ordering on ProjA. Let us recall that a C*-algebra B is monotone complete if each norm bounded, upward directed set in Bsa has a supremum in Bsa. Then, by considering approxi- mate units, it can be shown that B always has a unit element. (Another possible definition is: each upper bounded, upward directed set in Bsa has a supremum in Bsa. For unital algebras these are equivalent but for non-unital algebras they are not the same.) Kaplansky introduced AW*-algebras as an algebraic generalisation of von Neu- mann algebras (17).