Reconstructing Etale Groupoids from Semigroups

文献摘要

We unify various 'etale groupoid reconstruction theorems such as: Kumjian-Renault’s reconstruction from a groupoid C*-algebra. Exel’s reconstruction from an ample inverse semigroup. Steinberg’s reconstruction from a groupoid ring. Choi-Gardella-Thiel’s reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an 'etale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupoid G.

We unify various 'etale groupoid reconstruction theorems such as:

Kumjian-Renault’s reconstruction from a groupoid C*-algebra.
Exel’s reconstruction from an ample inverse semigroup.
Steinberg’s reconstruction from a groupoid ring.
Choi-Gardella-Thiel’s reconstruction from a groupoid L^p-algebra. We do this by working with certain bumpy semigroups S of functions defined on an 'etale groupoid G. The semigroup structure of S together with the diagonal subsemigroup D then yields a natural domination relation < on S. The groupoid of <-ultrafilters is then isomorphic to the original groupoid G.