Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



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
Published electronically: April 19, 2011
MathSciNet review: 2806681
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. P. Ara and E. Pardo, Refinement monoids with weak comparability and applications to regular rings and C$ ^*$-algebras. Proc. Amer. Math. Soc. 124 (1996), 715-720. MR 1301484 (96f:46124)
  • 2. P. Ara, K. R. Goodearl, K. C. O'Meara, and E. Pardo, Separative cancellation for projective modules over exchange rings. Israel J. Math. 105 (1998), 105-137. MR 1639739 (99g:16006)
  • 3. G. Aranda, K.R. Goodearl, F. Perera, and M. Siles Molina, Non-simple Purely Infinite Rings. Amer. J. Math. 132 (2010), no. 3, 563-610. MR 2666902
  • 4. L.G. Brown, Stable isomorphism of hereditary subalgebras of C$ ^*$-algebras. Pacific J. Math. 71 (1977), 335-348. MR 0454645 (56:12894)
  • 5. L. G. Brown and G. K. Pedersen, C$ ^*$-algebras of real rank zero. J. Funct. Anal. 99 (1991), 131-149. MR 1120918 (92m:46086)
  • 6. H. Dobbertin, Refinement monoids, Vaught monoids, and Boolean algebras. Math. Ann. 265 (1983), 473-487. MR 721882 (85e:06016)
  • 7. G. A. Elliott and D. Kucerovsky, An abstract Voiculescu-Brown-Douglas-Fillmore absorption theorem. Pacific J. Math. 198 (2001), 385-409. MR 1835515 (2002i:46052)
  • 8. I. Hirshberg, M. Rørdam, and W. Winter, $ C_0(X)$-algebras, stability and strongly self-absorbing C$ ^*$-algebras. Math. Ann. 339 (2007), 695-732. MR 2336064 (2008j:46040)
  • 9. J. Hjelmborg and M. Rørdam, On stability of C$ ^*$-algebras. J. Funct. Anal. 155 (1998), 153-171. MR 1623142 (99g:46079)
  • 10. D. Kucerovsky and P. W. Ng, $ S$-regularity and the corona factorization property. Math. Scand. 99 (2006), 204-216. MR 2289022 (2009g:46103)
  • 11. E. Ortega, F. Perera and M. Rørdam, The Corona Factorization Property, Stability, and the Cuntz semigroup of a $ C^*$-algebra. International Mathematics Research Notices (2011); doi: 10.1093/imrn/rnr013
  • 12. E. Pardo, Metric completions of ordered groups and $ K_0$ of exchange rings. Trans. Amer. Math. Soc. 350 (1998), 913-933. MR 1376552 (98e:46088)
  • 13. F. Perera, The structure of positive elements for C$ ^*$-algebras with real rank zero. Int. J. Math. 8 (1997), 383-405. MR 1454480 (98i:46058)
  • 14. F. Perera and M. Rørdam, AF-embeddings into C$ ^*$-algebras of real rank zero. J. Funct. Anal. 217 (2004), 142-170. MR 2097610 (2005g:46109)
  • 15. M. Pimsner, S. Popa, and D. Voiculescu, Homogeneous C$ ^*$-extensions of $ C(X)\otimes K(H)$, I. J. Operator Theory 1 (1979), 55-108. MR 526291 (82e:46093a)
  • 16. M. Rørdam, Stability of C$ ^*$-algebras is not a stable property. Doc. Math. 2 (1997), 375-386. MR 1490456 (98i:46060)
  • 17. M. Rørdam, A simple C$ ^*$-algebra with a finite and an infinite projection. Acta Math. 191 (2003), 109-142. MR 2020420 (2005m:46096)
  • 18. F. Wehrung, Injective positively ordered monoids, I. J. Pure Appl. Algebra 83 (1992), 43-82. MR 1190444 (93k:06023)
  • 19. F. Wehrung, Embedding simple commutative monoids into simple refinement monoids. Semigroup Forum 56 (1998), 104-129. MR 1490558 (99b:20092)
  • 20. S. Zhang, Matricial structure and homotopy type of simple C$ ^{*}$-algebras with real rank zero. J.Operator Theory 26 (1991), 283-312. MR 1225518 (94f:46075)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L35, 06F05, 46L80

Retrieve articles in all journals with MSC (2000): 46L35, 06F05, 46L80

Additional Information

Eduard Ortega
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

Francesc Perera
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Spain

Mikael Rørdam
Affiliation: Department of Mathematical Sciences, University of Copenhagen, Universitets- parken 5, DK-2100, Copenhagen Ø, Denmark

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

American Mathematical Society