C∗-algebras of real rank zero

Abstract

The concept of real rank of a C ∗-algebra is introduced as a non-commutative analogue of dimension. It is shown that real rank zero is equivalent to the previously defined conditions FS and HP, and that it is invariant under strong Morita equivalence, in particular under stable isomorphism. Real rank zero is also invariant under inductive limits and split extensions, and the class may well be regarded as the conceptual completion of the AF-algebras. In some cases, notably when the algebra is matroid, it is shown that the multiplier algebra also has real rank zero—although that is not true in general. By a result of G. J. Murphy, this implies a Weyl-von Neumann type result for self-adjoint multiplier elements in these cases.