Groupoid Semidirect Product Fell Bundles I- Actions by Isomorphisms

Abstract

Given an action of a groupoid by isomorphisms on a Fell bundle (over another groupoid), we form a semidirect-product Fell bundle, and prove that its C * -algebra is isomorphic to a crossed product.

Groupoid Semidirect Product Fell Bundles II- Principal Actions and Stabilization

“…Since p is open, we can lift { p ′ (t ′ i ) } using Fell's Criterion (see [HKQW,Lemma 2.1]). Thus, after passing to a subnet and relabeling, we can find t i → t in T such that p(t i ) = p ′ (t ′ i ).…”

“…Since we can replace h by h ′ and k by k ′ without changing the left-hand side of (4.1), we may as well assume s(h) = r(k) from the start. We can assume (see [HKQW,Remark 5.2]) that h, k, hk ∈ H u where u = ρ(s(h)) = ρ(r(k)), and where in turn ρ :…”

“…Example 4.6. Let H be the action groupoid for the right action of G on itself as in [HKQW,Example 5.6]. Then (x, y, xy) → y induces an isomorphism of G\H onto G.…”

“…Proposition 4.7. Suppose that H has a Haar system λ H , that G acts principally on H by isomorphisms, and that the action is invariant as in [HKQW,Definition 5.7]. Then G\H has a Haar system λ G\H given by…”

“…Proof. First, to see that p : G\A → G\H is an upper-semicontinuous Banach bundle, we need to verify axioms (B1)-(B4) of [HKQW,Definition 2.4]. This can be done almost exactly as in [KMRW98, Proposition 2.15] with minor alterations.…”