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

AbstractIt 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 Mn1⊕⋯⊕Mnr→A for some natural numbers r,n1,…,nr with nj⩾n for all j if and only if A has no representation of dimension less than n