AF-embeddings into C*-algebras of real rank zero

2003/10/21 by Francesc Perera, Perera, Francesc, Mikael Rordam +1
Mathematics · #06F05 (Secondary) #46L05 #46L35 (Primary) #46L80 #FOS: Mathematics #K-Theory and Homology (math.KT) #Operator Algebras (math.OA) #math.KT #math.OA #msc:06F05 #msc:46L05 #msc:46L35 #msc:46L80

paper · pdf · doi:10.48550/arxiv.math/0310340

28 pages

arxiv created 2003/10/21 · arxiv updated 2009/12/01

Abstract

It is proved that every separable C^*-algebra of real rank zero contains an AF-sub-C^*-algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two C^*-algebras and such that every projection in a matrix algebra over the large C^*-algebra is equivalent to a projection in a matrix algebra over the AF-sub-C^*-algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a C^*-algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital C^*-algebra A of real rank zero and a natural number n, then there is a unital ^*-homomorphism M_n1 ⊕ ... ⊕ M_nr → A for some natural numbers r,n₁, ...,n_r with n_j ≥ n for all j if and only if A has no representation of dimension less than n.

Related

• Classification of homomorphisms into simple Z-stable C^*-algebras
• Elements of C^*-algebras Attaining Their Norm in a Finite-Dimensional Representation
• Real rank of extensions of C*-algebras
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