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 monotone σ−complete.

On C*-Algebras Which Cannot Be Decomposed Into Tensor Products With Both Factors Infinite-Dimensional

“…By Corollary 2.4, von Neumann algebras and hence their continuous, linear images (such as the Calkin algebra) satisfy the hypothesis of Theorem 1.1. The assertion for AW*algebras follows from Proposition 2.5 as every maximal abelian self-adjoint subalgebra of an AW*-algebra is Grothendieck by the main results of [22] and [23]. The assertion (iii) follows from applying [19,Theorem 9] to the real Banach space A sa of all self-adjoint elements of a unital C*-algebra A with the countable Riesz interpolation property and noticing that the Grothendieck property passes from A sa to the complex Banach space A = A sa ⊕ iA sa .…”

𝐶*-algebras with weak* angelic dual balls

“…It is routine to show that there exists a maximal abelian * -subalgebra containing F (by using Zorn's lemma). We shall introduce (weakly) Rickart C * -algebras by using maximal abelian * -subalgebras; see [6,26] for the original definitions and their characterizations, and [27] for exposition of results on monotone (σ-)completeness.…”

Banach Synaptic Algebras

“…In [30], K. Saitô and J.D.M. Wright proved that a C * -algebra C is an AW * -algebra iff every maximal abelian * -subalgebra of C is monotone complete.…”