Vol. 112, No. 1, 1984

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ISSN: 0030-8730
On localizations and simple Cāˆ—-algebras

Alex Kumjian

Vol. 112 (1984), No. 1, 141ā€“192

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 G-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. K0(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.

Mathematical Subject Classification 2000
Primary: 46L55
Secondary: 46M20, 58H05
Received: 20 November 1981
Published: 1 May 1984
Alex Kumjian
University of Nevada