The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

Let A be a C*-algebra and I a closed two-sided ideal of A. We use the Hilbert C*-modules picture of the Cuntz semigroup to investigate the relations between the Cuntz semigroups of I, A and A/I. We obtain a relation on two elements of the Cuntz semigroup of A that characterizes when they are equal in the Cuntz semigroup of A/I. As a corollary, we show that the Cuntz semigroup functor is exact. Replacing the Cuntz equivalence relation of Hilbert modules by their isomorphism, w...

In the 1970s Alain Connes identified the appropriate notion of amenabilty for von Neumann algebras, and used it to obtain a deep internal finite dimensional approximation structure for these algebras. This structure is exactly what is needed for classification, and one of many consequences of Connes' theorem is the uniqueness of amenable II$_1$ factors, and later a complete classification of all simple amenable von Neumann algebras acting on separable Hilbert spaces. The El...

We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neu...

It is shown that, for a C*-algebra of stable rank one (i.e., in which the invertible elements are dense), two well-known isomorphism invariants, the Cuntz semigroup and the Thomsen semigroup, contain the same information. More precisely, these two invariants, viewed appropiately, determine each other in a natural way.

We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\cal Z}_0$ which has a unique tracial state and $K_0({\cal Z}_0)=\mathbb{Z}$ and $K_1({\cal Z}_0)=\{0\}.$ Let $A$ and $B$ be two separable simple $C^*$-algebras satisfying the UCT and have finite nuclear dimension. We show that $A\otimes {\cal Z}_0...

In this paper we describe the Cuntz semigroup of continuous fields of C$^*$-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yield...

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 ...

Cuntz and Li have defined a C*-algebra associated to any integral domain, using generators and relations, and proved that it is simple and purely infinite and that it is stably isomorphic to a crossed product of a commutative C*-algebra. We give an approach to a class of C*-algebras containing those studied by Cuntz and Li, using the general theory of C*-dynamical systems associated to certain semidirect product groups. Even for the special case of the Cuntz-Li algebras, our ...

Domain theory has its origins in Mathematics and Theoretical Computer Science. Mathematically it combines order and topology. Its central concepts have their origin in the idea of approximating ideal objects by their relatively finite or, more generally, relatively compact parts. The development of domain theory in recent years was mainly motivated by question in denotational semantics and the theory of computation. But since 2008, domain theoretical notions and methods are...

We introduce two nonnegative real-valued invariants for unital and stably finite C*-algebras whose minimal instances coincide with the notion of classifiability via the Elliott invariant. The first of these is defined for AH algebras, and may be thought of as a generalisation of slow dimension growth. The second invariant is defined for any unital and stably finite algebra, and may be thought of as an abstract version of the first invariant. We establish connections between b...