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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The Corona Factorization Property and refinement monoids
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by Eduard Ortega, Francesc Perera and Mikael Rørdam PDF
Trans. Amer. Math. Soc. 363 (2011), 4505-4525 Request permission

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
  • MR Author ID: 620835
  • 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
  • Received by editor(s): April 1, 2009
  • Published electronically: April 19, 2011
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2806681