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

Abstract

We unify the classic Dauns-Hofmann representation with Kumjian and Renault's Weyl groupoid representation. More precisely, we use ultrafilters to represent C*-algebras with some additional structure on Fell bundles over locally compact étale groupoids. Our construction is even functorial and thus a fully-fledged non-commutative extension of the classic Gelfand duality.

Representing Structured Semigroups on Étale Groupoid Bundles

“…Conversely, if θ is a faithful bundle representation then Theorem 11.2 yields a Pierce morphism (φ, τ) such that θ = (τ/φ) • C, where C is the coset bundle representation. As θ is injective, so is C, showing that (1) implies (2).…”

“…We also briefly examine ultrafilters in more general semigroups later in Theorem 12.8. Indeed, the present paper could be viewed as a prelude to our subsequent work in [2,3] where ultrafilters play a much greater role in noncommutative extensions of the classic Gelfand duality (see [13,14]).…”

“…In the present paper we have our eye on applications even further afield (see [2,3]), namely to algebraic analogues of C*-algebras known as Steinberg algebras (see [10,31]) as well as to noncommutative Cartan subalgebras of C*-algebras (see [12,19]). This forces us to consider more general semigroups S without an involution * , with more general noncommutative subsemigroups C. But we can always replace C with its centre…”

“…If S is the normaliser semigroup of a Cartan subalgebra C of some C*-algebra and N = Z = C then again we have principal filters a < , for each a ∈ S, which are also cosets, by Proposition 12.3 below. We also have principal filters in the more general structured C*-algebras considered in [2], thanks to [2, Proposition 6.11] and [3, Lemmas 5.2 and 5.3].…”

“…On the other hand, one might look for some natural restrictions or additional structures to place on structured semigroups to ensure that they do have faithful bundle representations. In our subsequent work (see [2,3]) we actually do a bit of both, considering semigroups that sit nicely within larger rings and their representations as sections of more general category bundles. In particular, if I ⊆ G is an ideal then g ∈ I implies gg −1 ∈ I and hence g −1 = g −1 gg −1 ∈ I, that is, I = I −1 = II.…”