The Corona Factorization Property and Refinement Monoids

TL;DR

This paper links the Corona Factorization Property of certain C*-algebras to a comparability property of their projection monoids, providing new insights into their structural characteristics and properties of projections.

Contribution

It establishes an equivalence between the Corona Factorization Property of algebras and a weak comparability property of their projection monoids, extending understanding of their structure.

Findings

A algebra has the Corona Factorization Property iff its projection monoid has a specific comparability property.

A projection is properly infinite iff a multiple of it is properly infinite in such algebras.

General results on conical refinement monoids are developed, including properties of order units and decompositions of properly infinite elements.

Abstract

The Corona Factorization Property of a C*-algebra, originally defined to study extensions of C*-algebras, has turned out to say something important about intrinsic structural properties of the C*-algebra. We show in this paper that a \sigma-unital C*-algebra A of real rank zero has the Corona Factorization roperty if and only if its monoid V(A) of Murray-von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C*-algebra is properly infinite if (and only if) a multiple of it is properly infinite. The latter result is obtained from some more general result we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite.