Marius Dadarlat

A universal multicoefficient theorem for the Kasparov groups

that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. There are advantages, in some circumstances, to using HomΛ(K(A),K(B)) in place of KK(A,B). These advantages derive from the fact that K(A) can be equipped with order and scale structures similar to those on K0(A). With this additional structure, the “Λ−module” K(A) becomes a powerful invariant of C*algebras. We show that it is a complete invariant for the class of real-rank-zero AD algebras. The AD algebras are a certain kind of approximately subhomogeneous C∗-algebras which may have torsion in K1 [Ell]. In addition to classifying these algebras, we calculate their automorphism groups up to approximately innerautomorphisms.

Constructions preserving Hilbert space uniform embeddability of discrete groups

Uniform embeddability (in a Hilbert space), introduced by Gromov, is a geometric property of metric spaces. As applied to countable discrete groups, it has important consequences for the Novikov conjecture. Exactness, introduced and studied extensively by Kirchberg and Wassermann, is a functional analytic property of locally compact groups. Recently it has become apparent that, as properties of countable discrete groups, uniform embeddability and exactness are closely related. We further develop the parallel between these classes by proving that the class of uniformly embeddable groups shares a number of permanence properties with the class of exact groups. In particular, we prove that it is closed under direct and free products (with and without amalgam), inductive limits and certain extensions.

Uniform embeddability of relatively hyperbolic groups

Abstract Let Γ be a finitely generated group which is hyperbolic relative to a finite family {H1, …,Hn } of subgroups. We prove that Γ is uniformly embeddable in a Hilbert space if and only if each subgroup Hi is uniformly embeddable in a Hilbert space.

Approximately unitarily equivalent morphisms and inductive limitC*-algebras

Let X be a compact metrizable space and let B be a unital C*-algebra. We prove that the .-homomorphisms from C(X) to B are classified up to stable approximate unitary equivalence by / ( t heo ry invariants. This result is used to obtain a classification theorem for certain real rank zero inductive limits of homogeneous C*-algebras. The classification of various classes of C*-algebras of real rank zero in terms of invariants based on K-theory presupposes a passage from algebraic objects to geometric objects (see jEll], [Lil], [EGLP1], [EG], [BrD], [G], [R0], [LiPh]). An underlying idea of this paper is that this passage can be done by using approximate morphisms. K-theory becomes a source of approximate morphisms thanks to the realization of K-theory in terms of asymptotic morphisms, due to A. Connes and N. Higson [CH]. The main results of the paper are Theorems A and B, below. By the universal coefficient theorem for the Kasparov KK-groups [RS], Ext(K.(C(X)), t(._I(B)) is a subgroup of Kt f (C(X) , B). Following [R0], we let KL(C(X) , B) denote the quotient of KK(C(X) , B) by the subgroup of pure extensions in Ext(K.(C(X)), K._I (B) ) . If K.(C(X)) is isomorphic to a direct sum of cyclic groups, then KL(C(X), B) coincides with KK(C(X) , B).

Asymptotic Unitary Equivalence in KK-Theory

A description of the Kasparov group KK(A,B) is given in terms of Cuntz pairs of representations and the notion of proper asymptotic unitary equivalence that we introduce here. The use of the word ‘proper’ reflects the crucial fact that all unitaries implementing the equivalence can be chosen to be compact perturbations of identity. The result has significant applications to the classification theory of nuclear C∗-algebras. Mathematics Subject Classifications (2000): 19K35, 46L80, 46L35.

ClassifyingC*-algebras via ordered, mod-p K-theory

We introduce an order structure on K 0 ( ) 9 K0(; Z/p ) . This group may also be thought of as Ko(; 7z @ Z/p ) . We exhibit new examples of real-rank zero C*-algebras that are inductive limits of finite dimensional and dimension-drop algebras, have the same ordered, graded K-theory with order unit and yet are not isomorphic. In fact they are not even stably shape equivalent. The order structure on K0(; Z ~3 Z / p ) naturally distinguishes these algebras. The same invariant is used to give an isomorphism theorem for such realrank zero inductive limits. As a corollary we obtain an isomorphism theorem for all real-rank zero approximately homogeneous C*-algebras that arise from systems of bounded dimension growth and torsion-free K0 group. At the 1980 Kingston conference, Effros posed the problem of finding suitable invariants for use in studying C*-algebras that are limits of sequences of homogeneous C*-algebras. These are now called almost homogeneous (AH) C*-algebras. The classification of AH algebras is a rapidly developing field and we will not attempt to summarize all this activity. Instead, we will focus on the growth of the invariants used. Specifically, we consider an AH algebra A that is the direct limit o f a system of the form

A Note on Asymptotic Homomorphisms

Using the notion of asymptotic homomorphism due to Connes and Higson we construct bivariant homology-cohomology theories for separable C*-algebras, which satisfy general excision axioms and are nonperiodic.

MORPHISMS OF SIMPLE TRACIALLY AF ALGEBRAS

Let A, B be separable simple unital tracially AF C*-algebras. Assuming that A is exact and satisfies the Universal Coefficient Theorem (UCT) in KK-theory, we prove the existence, and uniqueness modulo approximately inner automorphisms, of nuclear *-homomorphisms from A to B with prescribed K-theory data. This implies the AF-embeddability of separable exact residually finite-dimensional C*-algebras satisfying the UCT and reproves Huaxin Lins theorem on the classification of nuclear tracially AF C*-algebras.

E-theory for C⁎-algebras over topological spaces

Abstract We define E-theory for separable C ⁎ -algebras over second countable topological spaces and establish its basic properties. This includes an approximation theorem that relates the E-theory over a general space to the E-theories over finite approximations to this space. We obtain effective criteria for determining the invertibility of E-theory elements over possibly infinite-dimensional spaces. Furthermore, we prove a Universal Multicoefficient Theorem for C ⁎ -algebras over totally disconnected metrisable compact spaces.

Uniform embeddability and exactness of free products

Let A and B be countable discrete groups and let Γ=A∗B be their free product. We show that if both A and B are uniformly embeddable in a Hilbert space then so is Γ. We give two different proofs: the first directly constructs a uniform embedding of Γ from uniform embeddings of A and B; the second works without change to show that if both A and B are exact then so is Γ.