The Corona Factorization Property and refinement monoids

Ortega, Eduard; Perera, Francesc; Rørdam, Mikael

doi:10.1090/S0002-9947-2011-05480-2

The Corona Factorization Property and refinement monoids
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by Eduard Ortega, Francesc Perera and Mikael Rørdam

Trans. Amer. Math. Soc. 363 (2011), 4505-4525

DOI: https://doi.org/10.1090/S0002-9947-2011-05480-2

Published electronically: April 19, 2011

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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 Property if and only if its monoid $\mathrm V(A)$ 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 results that 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.

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