Non-commutative Stone duality: inverse semigroups, topological groupoids and C*-algebras

We study a non-commutative generalization of Stone duality that connects a class of inverse semigroups, called Boolean inverse $\wedge$-semigroups, with a class of topological groupoids, called Hausdorff Boolean groupoids. Much of the paper is given over to showing that Boolean inverse $\wedge$-semigroups arise as completions of inverse semigroups we call pre-Boolean. An inverse $\wedge$-semigroup is pre-Boolean if and only if every tight filter is an ultrafilter, where the definition of a tight filter is obtained by combining work of both Exel and Lenz. A simple necessary condition for a semigroup to be pre-Boolean is derived and a variety of examples of inverse semigroups are shown to satisfy it. Thus the polycyclic inverse monoids, and certain Rees matrix semigroups over the polycyclics, are pre-Boolean and it is proved that the groups of units of their completions are precisely the Thompson-Higman groups $G_{n,r}$. The inverse semigroups arising from suitable directed graphs are also pre-Boolean and the topological groupoids arising from these graph inverse semigroups under our non-commutative Stone duality are the groupoids that arise from the Cuntz-Krieger $C^{\ast}$-algebras.Comment: The presentation has been sharpened up and some minor errors correcte

Sets of lengths in maximal orders in central simple algebras

Let $\mathcal O$ be a holomorphy ring in a global field $K$, and $R$ a classical maximal $\mathcal O$-order in a central simple algebra over $K$. We study sets of lengths of factorizations of cancellative elements of $R$ into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of $\mathcal O$, which implies that all the structural finiteness results for sets of lengths---valid for commutative Krull monoids with finite class group---hold also true for $R$. If $\mathcal O$ is the ring of algebraic integers of a number field $K$, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.Comment: 40 pages; final version, with minor edits over previous on

Factorizations of Elements in Noncommutative Rings: A Survey

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of Chatters and Jordan and generalizations thereof. We recall arithmetical invariants for the study of non-unique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.Comment: 50 pages, comments welcom

Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-indexed commutative diagram D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.Comment: 34 pages. Algebras and Representation Theory, to appea