On localizations and simpleC^∗-algebras

Abstract

A method for associating C*-algebras to inverse semigroups of partial homeomorphisms (termed localizations) is developed. Localizations which locally have the same structure yield C*-algebras in the same strong Morita equivalence class (via the linking algebra characterization).Free localizations are closely related to Renault's principal discrete groupoids, where the partial homeomorphisms are identified with open (7-sets. The space on which a free localization is defined becomes the spectrum of a "Cartan" masa in the associated C*-algebra (but this masa is not unique modulo conjugation by automorphisms).It is shown that if A is a simple unital AF algebra with comparability of projections (i.e. K 0 (A) can be embedded as an ordered subgroup of the reals) then A embeds unitally in the transformation-group algebra associated to the action of a discrete subgroup of the unit circle.

Regular representation of groupoid C* -algebras and applications to inverse semigroups

“…the pair ðX ; SÞ is a localization in the sense of [12], [16]) in the following manner. If x A F u à u , let u:x be the character e 7 !…”

On a Groupoid Construction for Actions of Certain Inverse Semigroups

“…The point of view of this note is slightly different from the one of [6]; for instance, the approach to the Glimm groupoid mentioned in Remark 4.2 above is different from the one taken in [6] (see 5.2 below); actually, the inverse semigroup generated by the restricted odometer map is not a localization, and so is also the case for the example described in Section 3 (even in the classical situation when (G, P ) = (Z, N) ).…”

“…Still, the C*-algebra construction of [6] coincides with the C*-algebra of the groupoid defined in 2.2, on the common territory of the two approaches:…”

Examples of different minimal diffeomorphisms giving the same C*-algebras

“…For minimal homeomorphisms of the Cantor set, a remarkable theorem of Giordano, Putnam and Skau (Theorem 2.1 of [10]) asserts that isomorphism of of the introduction to [13] for nonisomorphic Cartan subalgebras, and see Theorem 1.13 below, which is immediate from results already in the literature, for isomorphic but nonconjugate Cartan subalgebras. Our examples give both simple nuclear C*-algebras with new kinds of isomorphic but nonconjugate Cartan subalgebras, and simple nuclear C*-algebras with many Cartan subalgebras with quite different structure, being algebras of continuous functions on compact spaces of different dimensions.…”