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The Corona Factorization Property and refinement monoids


Authors: Eduard Ortega, Francesc Perera and Mikael Rørdam
Journal: Trans. Amer. Math. Soc. 363 (2011), 4505-4525
MSC (2000): Primary 46L35, 06F05; Secondary 46L80
DOI: https://doi.org/10.1090/S0002-9947-2011-05480-2
Published electronically: April 19, 2011
MathSciNet review: 2806681
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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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Additional Information

Eduard Ortega
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
Email: Eduardo.Ortega@math.ntnu.no

Francesc Perera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Spain
Email: perera@mat.uab.cat

Mikael Rørdam
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, DK-2100, Copenhagen Ø, Denmark
Email: rordam@math.ku.dk

DOI: https://doi.org/10.1090/S0002-9947-2011-05480-2
Keywords: C$^{*}$-algebras, Corona Factorization Property, real rank zero, conical refinement monoids
Received by editor(s): April 1, 2009
Published electronically: April 19, 2011
Article copyright: © Copyright 2011 American Mathematical Society

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