Abstract:
It is proved that if X is a compact Hausdorff space of Lebesgue dimension $\dim(X)$, then the squaring mapping $\alpha_{m} \colon (C(X)_{\mathrm{sa}})^{m} \to C(X)_{+}$, defined by $\alpha_{m}(f_{1},..., f_{m}) = \sum_{i=1}^{m} f_{i}^{2}$, is open if and only if $m -1 \ge \dim(X)$. Hence the Lebesgue dimension of X can be detected from openness of the squaring maps $\alpha_m$. In the case m=1 it is proved that the map $x \mapsto x^2$, from the self-adjoint elements of a unital $C^{\ast}$-algebra A into its positive elements, is open if and only if A is isomorphic to C(X) for some compact Hausdorff space X with $\dim(X)=0$.

Abstract:
An example is given of a simple, unital C*-algebra which contains an infinite and a non-zero finite projection. This C*-algebra is also an example of an infinite simple C*-algebra which is not purely infinite. A corner of this C*-algebra is a finite, simple, unital C*-algebra which is not stably finite. Our example shows that the type decomposition for von Neumann factors does not carry over to simple C*-algebras. Added March 2002: We also give an example of a simple, separable, nuclear C*-algebra in the UCT class which contains an infinite and a non-zero finite projection. This nuclear C*-algebra arises as a crossed product of an inductive limit of type I C*-algebras by an action of the integers.

Abstract:
We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\infty tensorially. Thus one can reach an O_\infty-absorbing C*-algebra as an inductive limit of the finite and elementary C*-algebras C_0([0,1),M_k). As an application we give a new proof of a recent theorem of Ozawa that the cone over any separable exact C*-algebra is AF-embeddable, and we exhibit a concrete AF-algebra into which this class of C*-algebras can be embedded.

Abstract:
Suppose that A is a C*-algebra for which A is isomorphic to A tensor Z, where Z is the Jiang-Su algebra: a unital, simple, stably finite, separable, nuclear, infinite dimensional C*-algebra with the same Elliott invariant as the complex numbers. We show that: (i) The Cuntz semigroup W(A) of equivalence classes of positive elements in matrix algebras over A is weakly unperforated. (ii) If A is exact, then A is purely infinite if and only if A is traceless. (iii) If A is separable and nuclear, then A is isomorphic to A tensor O_infty if and only if A is traceless. (iv) If A is simple and unital, then the stable rank of A is one if and only if A is finite. We also characterise when A is of real rank zero.

Abstract:
We show that if A is a separable, nuclear, O_infty-absorbing (or strongly purely infinite) C*-algebra, which is homotopic to zero in an ideal-system preserving way, then A is the inductive limit of C*-algebras of the form M_k(C_0(G,v)), where G is a finite graph (and C_0(G,v) is the algebra of continuous functions on G that vanish at a distinguished point v in G). We show further that any separable, nuclear, stable, O_2-absorbing C*-algebra is isomorphic to a crossed product of a C*-algebra D with the integers by an action alpha, where D is an inductive limit of C*-algebras of the form M_k(C_0(G,v)) (and D is O_2-absorbing and homotopic to zero in an ideal-system preserving way).

Abstract:
We give a number of new characterizations of the Jiang-Su algebra Z, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the way we study divisibility properties of C*-algebras, we give a precise characterization of those unital C*-algebras of stable rank one that admit a unital embedding of the dimension-drop C*-algebra Z_{n,n+1}, and we prove a cancellation theorem for the Cuntz semigroup of C*-algebras of stable rank one.

Abstract:
We study conditions that will ensure that a crossed product of a C*-algebra by a discrete exact group is purely infinite (simple or non-simple). We are particularly interested in the case of a discrete non-amenable exact group acting on a commutative C*-algebra, where our sufficient conditions can be phrased in terms of paradoxicality of subsets of the spectrum of the abelian C*-algebra. As an application of our results we show that every discrete countable non-amenable exact group admits a free amenable minimal action on the Cantor set such that the corresponding crossed product C*-algebra is a Kirchberg algebra in the UCT class.

Abstract:
We study properties of the central sequence algebra of a C*-algebra, and we present an alternative approach to a recent result of Matui and Sato. They prove that every unital separable simple nuclear C*-algebra, whose trace simplex is finite dimensional, tensorially absorbs the Jiang-Su algebra if and only if it has the strict comparison property. We extend their result to the case where the extreme boundary of the trace simplex is closed and of finite topological dimension. We are also able to relax the assumption on the C*-algebra of having the strict comparison property to a weaker property, that we call local weak comparison. Namely, we prove that a unital separable simple nuclear C*-algebra, whose trace simplex has finite dimensional closed extreme boundary, tensorially absorbs the Jiang-Su algebra if and only if it has the local weak comparison property. We can also eliminate the nuclearity assumption, if instead we assume the (SI) property of Matui and Sato, and, moreover, that each II_1-factor representation of the C*-algebra is a McDuff factor.

Abstract:
We study universal properties of locally compact G-spaces for countable infinite groups G. In particular we consider open invariant subsets of the \beta-compactification of G (which is a G-space in a natural way), and their minimal closed invariant subspaces. These are locally compact free G-spaces, and the latter are also minimal. We examine the properies of these G-spaces with emphasis on their universal properties. As an example of our resuts, we use combinatorial methods to show that each countable infinite group admits a free minimal action on the locally compact non-compact Cantor set.

Abstract:
It is shown that a strongly self-absorbing C*-algebra is of real rank zero and absorbs the Jiang-Su algebra if it contains a nontrivial projection. We also consider cases where the UCT is automatic for strongly self-absorbing C*-algebras, and K-theoretical ways of characterizing when Kirchberg algebras are strongly self-absorbing.