N. Christopher Phillips

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 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 is a separable C*-algebra whose irreducible representations all have dimension at most and if for each the space of -dimensional irreducible representations has finite covering dimension, then is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.

Cancellation and stable rank for direct limits of recursive subhomogeneous algebras

We prove the following results for a unital simple direct limit of recursive subhomogeneous algebras with no dimension growth: (1) (2) The projections in satisfy cancellation: if then (3) satisfies Blackadars Second Fundamental Comparability Question: if are projections such that for all normalized traces on then (4) is unperforated for the strict order: if and there is such that then The last three of these results hold under certain weaker dimension growth conditions and without assuming simplicity. We use these results to obtain previously unknown information on the ordered K-theory of the crossed product obtained from a minimal homeomorphism of a finite-dimensional infinite compact metric space Specifically, is unperforated for the strict order, and satisfies the following K-theoretic version of Blackadars Second Fundamental Comparability Question: if satisfies for all normalized traces on then there is a projection such that --------------------------------------------------------------------------------

Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism also has these three properties. We also present examples of C*-algebras and automorphisms which satisfy the above assumptions, but such that crossed product algebras do not have tracial rank zero.

The Calkin algebra has outer automorphisms

Assuming the continuum hypothesis, we show that the Calkin algebra has 2^{aleph_1} outer automorphisms.

Ranks of Algebras of Continuous C -Algebra Valued Functions

We prove a number of results about the stable and particularly the real ranks of tensor products of C � -algebras under the assumption that one of the factors is commutative. In particular, we prove the following:

Analogs of Cuntz algebras on Lp spaces

For

Banach algebras and rational homotopy theory

d = 2, 3, \ldots

Representable -theory of smooth crossed products by and

and

Permanence properties for crossed products and fixed point algebras of finite groups

p \in [1, \infty),

A direct proof of -stability for approximately homogeneous C*-algebras of bounded topological dimension

we define a class of representations