K0 of multiplier algebras of C*-algebras with real rank zero

For a large class of -unital C ^*-algebras A with real rank zero and stable rank one, the structure of the Grothendieck group k ₀ of the multiplier algebra (A) is investigated. The ordered group K ₀( (A)) is shown to be an unperforated Riesz group, and its additive structure is completely determined, as is — in important cases — its order structure. These structures, and the attendant consequences for the ideal structure of (A), are richer than previously anticipated. In particular, it is shown that the corona algebra (A)/A can have very large stably finite quotient algebras. For example, there exist simple, separable, approximately finite-dimensional C ^*-algebras A such that the maximal stably finite quotient algebra of (A)/A has uncountably many maximal ideals modulo which a W ^*-factor of Type II₁ results. The analysis of the additive structure of K ₀( (A)) yields as a byproduct that if A is a -unital approximately finite-dimensional C ^*-algebra without nonzero unital quotient algebras, then all quasitraces on (A) are traces.This research was partially supported by a grant from the National Science Foundation.     

K-theory and exact sequences of partial translation algebras

In an earlier paper, the authors introduced partial translation algebras as a generalisation of group C^?${\mathrm{C}}^{?}$-algebras. Here we establish an extension of partial translation algebras, which may be viewed as an excision theorem in this context. We apply this general framework to compute the K-theory of partial translation algebras and group C^?${\mathrm{C}}^{?}$-algebras in the context of almost invariant subspaces of discrete groups. This generalises the work of Cuntz, Lance, Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's calculation of the K-theory for a graph product of groups.     

UCT-Kirchberg algebras have nuclear dimension one

We prove that every Kirchberg algebra in the UCT class has nuclear dimension 1. We first show that Kirchberg 2-graph algebras with trivial _K0$K_0$ and finite _K1$K_1$ have nuclear dimension 1 by adapting a technique developed by Winter and Zacharias for Cuntz algebras. We then prove that every Kirchberg algebra in the UCT class is a direct limit of 2-graph algebras to obtain our main theorem.     

Bivariant K-theory via correspondences

We use correspondences to define a purely topological equivariant bivariant K-theory for spaces with a proper groupoid action. Our notion of correspondence differs slightly from that of Connes and Skandalis. Our construction uses no special features of equivariant K-theory. To highlight this, we construct bivariant extensions for arbitrary equivariant multiplicative cohomology theories.We formulate necessary and sufficient conditions for certain duality isomorphisms in the topological bivariant K-theory and verify these conditions in some cases, including smooth manifolds with a smooth cocompact action of a Lie group. One of these duality isomorphisms reduces bivariant K-theory to K-theory with support conditions. Since similar duality isomorphisms exist in Kasparov theory, the topological and analytic bivariant K-theories agree if there is such a duality isomorphism.     

Universally generic finitely generated ordered Abelian groups

It is shown that a finitely generated ordered Abelian group is generic if and only if it is superdiscrete, i.e., each homomorphic image is discretely ordered. The forcing concept uses universal sentences as forcing conditions.     

The local index formula in semifinite Von Neumann algebras I: Spectral flow

We generalise the local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. In this setting it gives a formula for spectral flow along a path joining an unbounded self-adjoint Breuer-Fredholm operator, affiliated to the von Neumann algebra, to a unitarily equivalent operator. Our proof is novel even in the setting of the original theorem and relies on the introduction of a function valued cocycle which is ‘almost’ a (b,B)-cocycle in the cyclic cohomology of A.     

Topological groups and covering dimension

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional topological groups. The dimension function that we introduce extends Lebesgue covering dimension, has the hereditary property, and has a product theory that is more similar to the product theory for the finite dimensional case.     

Orbifold index cobordism invariance

We prove cobordism index invariance for pseudo-differential elliptic operators on closed orbifolds with K-theoretical methods.     

The local index formula in semifinite von Neumann algebras II: The even case

We generalise the even local index formula of Connes and Moscovici to the case of spectral triples for a *-subalgebra A of a general semifinite von Neumann algebra. The proof is a variant of that for the odd case which appears in Part I. To allow for algebras with a non-trivial centre we have to establish a theory of unbounded Fredholm operators in a general semifinite von Neumann algebra and in particular prove a generalised McKean-Singer formula.     

The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras

The probability measures on compact Hausdorff spaces K form a compact convex subset PK of the space of measures with the vague topology. Every continuous map of compact Hausdorff spaces induces a continuous affine map extending f. Together with the canonical embedding associating to every point its Dirac measure and the barycentric map β associating to every probability measure on PK its barycenter, we obtain a monad (P,ε,β). The Eilenberg-Moore algebras of this monad have been characterised to be the compact convex sets embeddable in locally convex topological vector spaces by Swirszcz [T. Swirszcz, Monadic functors and convexity, Bul. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 22 (1974) 39-42].We generalise this result to compact ordered spaces in the sense of Nachbin [L. Nachbin, Topology and Order, Von Nostrand, Princeton, NJ, 1965. Translated from the 1950 monograph “Topologia e Ordem” (in Portugese). Reprinted by Robert E. Kreiger Publishing Co., Huntington, NY, 1967]. The probability measures form again a compact ordered space when endowed with the stochastic order. The maps ε and β are shown to preserve the stochastic orders. Thus, we obtain a monad over the category of compact ordered spaces and order preserving continuous maps. The algebras of this monad are shown to be the compact convex ordered sets embeddable in locally convex ordered topological vector spaces.This result can be seen as a step towards the characterisation of the algebras of the monad of probability measures on the category of stably compact spaces (see [G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices and Domains, Encyclopedia Math. Appl., vol. 93, Cambridge University Press, 2003, Section VI-6]).     

Boundary amenability of relatively hyperbolic groups

Let K be a fine hyperbolic graph and Γ be a group acting on K with finite quotient. We prove that Γ is exact provided that all vertex stabilizers are exact. In particular, a relatively hyperbolic group is exact if all its peripheral groups are exact. We prove this by showing that the group Γ acts amenably on a compact topological space. We include some applications to the theories of group von Neumann algebras and of measurable orbit equivalence relations.     

On the Kurosh rank of the intersection of subgroups in free products of groups

The Kurosh rank rK(H) of a subgroup H of a free product of groups Gα, α∈I, is defined accordingly to the classic Kurosh subgroup theorem as the number of free factors of H. We prove that if H₁, H₂ are subgroups of and H₁, H₂ have finite Kurosh rank, then , where , q^∗ is the minimum of orders >2 of finite subgroups of groups Gα, α∈I, q^∗:=∞ if there are no such subgroups, and if q^∗=∞. In particular, if the factors Gα, α∈I, are torsion-free groups, then .     

Cancellation and absorption of lexicographic powers of totally ordered Abelian groups

Let G and H be totally ordered Abelian groups such that, for some integer k, the lexicographic powers G ^k and H ^k are isomorphic (as ordered groups). It was proved by F. Oger that G and H need not be isomorphic. We show here that they are whenever G is either divisible or ₁ -saturated (and in a few more cases). Our proof relies on a general technique which we also use to prove that G and H must be elementary equivalent as ordered groups (a fact also proved by F. Delon and F. Lucas) and isomorphic as chains. The same technique applies to the question of whether G and H should be isomorphic as groups, but, in this last case, no hint about a possible negative answer seems available.     

Banach algebras on compact right topological groups

It is shown how the basic constructs of harmonic analysis, such as convolution, algebras of measures and functions (including Fourier-Stieltjes algebras) can be developed for compact Hausdorff right topological groups. In particular, the properties and structure of these new objects are compared with their classical analogues in the topological group case.     

Continuous crystal and Duistermaat-Heckman measure for Coxeter groups

We introduce a notion of continuous crystal analogous, for general Coxeter groups, to the combinatorial crystals introduced by Kashiwara in representation theory of Lie algebras. We explore their main properties in the case of finite Coxeter groups, where we use a generalization of the Littelmann path model to show the existence of the crystals. We introduce a remarkable measure, analogous to the Duistermaat-Heckman measure, which we interpret in terms of Brownian motion. We also show that the Littelmann path operators can be derived from simple considerations on Sturm-Liouville equations.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

Chainability and Hemmingsen's theorem

On the surface, the definitions of chainability and Lebesgue covering dimension ?1 are quite similar as covering properties. Using the ultracoproduct construction for compact Hausdorff spaces, we explore the assertion that the similarity is only skin deep. In the case of dimension, there is a theorem of E. Hemmingsen that gives us a first-order lattice-theoretic characterization. We show that no such characterization is possible for chainability, by proving that if κ is any infinite cardinal and A is a lattice base for a nondegenerate continuum, then A is elementarily equivalent to a lattice base for a continuum Y, of weight κ, such that Y has a 3-set open cover admitting no chain open refinement.     

Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements

This paper axiomatizes classes of Abelian lattice-ordered groups with a finite upper bound on the number of pairwise disjoint positive elements; finds model-completions for these theories; derives corresponding Nullstellensätze; determines which model-completions eliminate quantifiers; and examines quantifier elimination in a different language and for positive formulas.