AF-embeddings into [formula omitted]-algebras of real rank zero

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 n 1 ⊕⋯⊕ M n r → A for some natural numbers r, n 1,…, n r with n j ⩾ n for all j if and only if A has no representation of dimension less than n.