Real rank and property (SP) for direct limits of recursive subhomogeneous algebras

Let A be a unital simple direct limit of recursive subhomogeneous C*-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 natural map from K_0 (A) to the continuous affine functions on the tracial state space has dense range, A has the Property (SP) if and only if the range of this map contains strictly positive functions with arbitrarily small norm. By comparison with results for unital simple direct limit of homogeneous C*-algebras with no dimension growth, one might hope that weaker conditions might suffice. We give examples to show that several plausible weaker conditions do not suffice for the results above. If A has real rank zero and at most countably many extreme tracial states, we apply results of H. Lin to show that A has tracial rank zero and is classifiable.Comment: 23 pages, AMSLaTe

Crossed products by finite cyclic group actions with the tracial Rokhlin property

We define the tracial Rokhlin property for actions of finite cyclic groups on stably finite simple unital C*-algebras. We prove that when the algebra is in addition simple and has tracial rank zero, then the crossed product again has tracial rank zero. Under a kind of weak approximate innerness assumption and one other technical condition, we prove that if the action has the the tracial Rokhlin property and the crossed product has tracial rank zero, then the original algebra has tracial rank zero. We give examples showing how the tracial Rokhlin property differs from the Rokhlin property of Izumi. We use these results, together with work of Elliott-Evans and Kishimoto, to give an inductive proof that every simple higher dimensional noncommutative torus is an AT algebra. We further prove that the crossed product of every simple higher dimensional noncommutative torus by the flip is an AF algebra, and that the crossed products of irrational rotation algebras by the standard actions of the cyclic groups of orders 3, 4, and 6 are simple AH algebras with real rank zero.Comment: 90 pages, AMSLaTe

The tracial Rokhlin property is generic

We prove several results of the following general form: automorphisms of (or actions of ${\mathbb{Z}}^d$ on) certain kinds of simple separable unital C*-algebras $A$ which have a suitable version of the Rokhlin property are generic among all automorphisms (or actions), or in a suitable class of automorphisms. That is, the ones with the version of the Rokhlin property contain a dense $G_{\delta}$-subset of the set of all such automorphisms (or actions). Specifically, we prove the following. If $A$ is stable under tensoring with the Jiang-Su algebra $Z,$ and has tracial rank zero, then automorphisms with the tracial Rokhlin property are generic. If $A$ has tracial rank zero, or, more generally, $A$ is tracially approximately divisible together with a technical condition, then automorphisms with the tracial Rokhlin property are generic among the approximately inner automorphisms. If $A$ is stable under tensoring with the Cuntz algebra ${\mathcal{O}}_{\infty}$ or with a UHF algebra of infinite type, then actions of ${\mathbb{Z}}^d$ on $A$ with the Rokhlin property are generic among all actions of ${\mathbb{Z}}^d.$ We further give a related but more restricted result for actions of finite groups.Comment: AMSLaTeX, 33 page

Large subalgebras

We define and study large and stably large subalgebras of simple unital C*-algebras. The basic example is the orbit breaking subalgebra of a crossed product by Z, as follows. Let X be an infinite compact metric space, let h be a minimal homeomorphism of X, and let Y be a closed subset of X. Let u be the standard unitary in C* (Z, X, h). The Y-orbit breaking subalgebra is the subalgebra of C* (Z, X, h) generated by C (X) and all elements f u for f in C (X) such that f vanishes on Y. If intersects each orbit of h at most once, then the Y-orbit breaking subalgebra is large in C* (Z, X, h). Large subalgebras obtained via generalizations of this construction have appeared in a number of places, and we unify their theory in this paper. We prove the following results for an infinite dimensional simple unital C*-algebra A and a stably large subalgebra B of A: B is simple and infinite dimensional. If B is stably finite then so is A, and if B is purely infinite then so is A. The restriction maps from the tracial states of A to the tracial states of B and from the normalized 2-quasitraces on A to the normalized 2-quasitraces on B are bijective. When A is stably finite, the inclusion of B in A induces an isomorphism on the semigroups that remain after deleting from the Cuntz semigroups of A and B all the classes of nonzero projections. B and A have the same radius of comparison.Comment: 54 pages; AMSLaTe

Crossed products of the Cantor set by free minimal actions of Z^d

Let d be a positive integer, let X be the Cantor set, and let Z^d act freely and minimally on X. We prove that the crossed product C* (Z^d, X) has stable rank one, real rank zero, and cancellation of projections, and that the order on K_0 (C* (Z^d, X)) is determined by traces. We obtain the same conclusion for the C*-algebras of various kinds of aperiodic tilings.Comment: 41 pages, AMSLaTe

Examples of different minimal diffeomorphisms giving the same C*-algebras

We give examples of minimal diffeomorphisms of compact connected manifolds which are not topologically orbit equivalent, but whose transformation group C*-algebras are isomorphic. The examples show that the following properties of a minimal diffeomorphism are not invariants of the transformation group C*-algebra: having topologically quasidiscrete spectrum; the action on singular cohomology (when the manifolds are diffeomorphic); the homotopy type of the manifold (when the manifolds have the same dimension); and the dimension of the manifold. These examples also give examples of nonconjugate isomorphic Cartan subalgebras, and of nonisomorphic Cartan subalgebras, of simple separable nuclear unital C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem.Comment: AMSLaTeX; 21 pages, no figure

Recursive subhomogeneous algebras

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form C (X, M_n) many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies in particular that if A is a separable C*-algebra whose irreducible representations all have dimension at most N (for some finite N), and if for each n the space of n-dimensional irreducible representations has finite covering dimension, then A is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it.Comment: 29 pages, AMSLaTe

When are crossed products by minimal diffeomorphisms isomorphic?

This is a survey which discusses the isomorphism problem for both C* and smooth crossed products by minimal diffeomorphisms. For C* crossed products, examples demonstrate the failure of the obvious analog of the Giordano-Putnam-Skau Theorem on minimal homeomorphisms of the Cantor set. For smooth crossed products, there are many open problems.Comment: 23 pages, AMSLaTeX. To appear in the proceedings of the Conference on Operator Algebras and Mathematical Physics, Constanta, Romania (2001

Every simple higher dimensional noncommutative torus is an AT algebra

We prove that every simple higher dimensional noncommutative torus is an AT algebra.Comment: AMSLaTeX; 22 pages. This paper replaces Sections 5 through 7 of the unpublished long preprint arXiv:math.OA/0306410. A number of minor improvements have been made, particularly near the en

The C*-algebra of a minimal homeomorphism with finite mean dimension has finite radius of comparison

Let $X$ be an infinite compact metric space and let $h$ be a minimal homeomorphism of $X$. We prove that the radius of comparison of the transformation group C*-algebra of $h$ is at most $1$ plus $36$ times the mean dimension of $h$.Comment: 36 pages; AMSLaTe