Let A be a unital simple direct limit of recursive subhomogeneous algebras with no dimension growth. We give criteria which specify exactly when A has real rank zero, and exactly when A has the Property (SP): every nonzero hereditary subalgebra of A contains a nonzero projection. Specifically, A has real rank zero if and only if the image of K0(A) in Aff(T (A)) is dense, and A has the Property ...

We introduce a general class of automorphisms of rotation algebras, the noncommutative Furstenberg transformations. We prove that fully irrational noncommutative Furstenberg transformations have the tracial Rokhlin property, which is a strong form of outerness. We conclude that crossed products by these automorphisms have stable rank one, real rank zero, and order on projections determined by t...

We describe a class of rank-2 graphs whose C∗-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C∗-algebra. We identify rank-2 Bratteli diagrams whose C∗-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C∗-algebras contain as f...

The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i 6= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper surveys the current state of knowledge on the problem of determining the minimum rank of a graph and related issues.