arXiv subjects
Publications and source records attributed to George A. Elliott.
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 report on recent progress in the program to classify separable amenable C*-algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.
A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.
A classification is given of certain separable nuclear C*-algebras not necessarily of real rank zero, namely, the class of separable simple C*-algebras which are inductive limits of continuous-trace C*-algebras whose building blocks have spectrum homeomorphic to the closed interval [0,1], or to a disjoint union of copies of this space. Also, the range of the invariant is calculated.
We study the limits of inductive sequences (A_i,ϕ_i) where each A_i is a direct sum of full matrix algebras over compact metric spaces and each partial map of ϕ_i is diagonal. We give a new characterisation of simplicity for such algebras, and apply it to prove that the said algebras have stable rank one whenever they are simple and unital. Significantly, our results do not require any dimension growth assumption.
We show that in the generic case the smooth noncommutative tori associated to two n by n real skew-symmetric matrices are Morita equivalent if and only if the matrices are in the same orbit of the natural SO(n, n| Z) action.
We investigate the class of unital C*-algebras that admit a unital embedding into every unital C*-algebra of real rank zero, that has no finite-dimensional quotients. We refer to a C*-algebra in this class as an initial object. We show that there are many initial objects, including for example some unital, simple, infinite-dimensional AF-algebras, the Jiang-Su algebra, and the GICAR-algebra. That the GICAR-algebra is an initial object follows from an analysis of Hausdorff moment sequences. It is shown that a dense set of Hausdorff moment sequences belong to a given dense subgroup of the real numbers, and hence that the Hausdorff moment problem can be solved (in a non-trivial way) when the moments are required to belong to an arbitrary simple dimension group (i.e., unperforated simple ordered group with the Riesz decomposition property).
We show that two C*-algebraic noncommutative tori are strongly Morita equivalent if and only if they have isomorphic ordered K_0-groups and centers, extending N. C. Phillips's result in the case that the algebras are simple. This is also generalized to the twisted group C*-algebras of arbitrary finitely generated abelian groups.