Representing Structured Semigroups on Etale Groupoid Bundles

Abstract

We examine a semigroup analogue of the Kumjian-Renault representation of C*-algebras with Cartan subalgebras on twisted groupoids. Specifically, we show how to represent semigroups with distinguished normal subsemigroups as 'slice-sections' of étale groupoid bundles.

Dauns-Hofmann-Kumjian-Renault Duality for Fell Bundles and Structured C*-Algebras

“…This harks back to classic topological dualities due to Stone [Sto36], Wallman [Wal38] and Milgram [Mil49], and is in the same spirit as other non-commutative dualities obtained more recently in [Law12], [LL13], [KL16] and [BS19]. Essentially the same construction even applies in more general algebraic contexts, as we have examined in [Bic20b] and [Bic20a]. Indeed, the present paper builds on and could be viewed as the culmination of this previous work.…”

“…Here we examine several interrelated properties of A which will be important for our later work. Many proofs make use of the continuous functional calculus/Gelfand representation, something we did not have access too in our algebraic predecessors [Bic20b] and [Bic20a]. In particular, we will often need a sequence of polynomials (f n ) that is bounded on bounded subsets of R, approaches 1 uniformly on compact subsets of R \ {0} and such that f n (0) = 0, for all n (such a sequence exists thanks to Stone-Weierstrass).…”

“…Diagonality turned out to be needed in [Bic20b] for parts of the semigroup theory relating to filters and cosets. Here we have the following general result.…”

“…In general, this is no longer true and, in order to use the theory in [Bic20b] and [Bic20a], we will need to separately assume that Z or its positive unit ball is binormal.…”

“…Structured C*-algebras consisting of sections of Fell bundles are indeed wellstructured, thanks to Proposition 5.23. Well-structured C*-algebras also satisfy the key assumptions required in [Bic20b] and [Bic20a], allowing us to make free use of the theory developed there. For example, N in [Bic20b, §3], which corresponds to our Φ[S], is required to be 'Z 1 + -trinormal', but this holds in well-structured C*-structures because Φ is normal/shiftable -see [Bic20a, Proposition 1.6].…”