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.
