Twisted Steinberg algebras

Reconstruction of Twisted Steinberg Algebras

“…Discrete twists. We next recall [4,Definition 4.1]. Let G be an étale groupoid, let R be a commutative unital ring, and let T be a subgroup of R ⇥ .…”

“…Thus ⌃ is étale by the previous assertion. Now [4,Lemma 4.3(b)] (which formally assumes that ⌃ and G are Hausdor↵ but only uses that they are étale and that conditions (DT1)-(DT3) are satisfied) implies that i is open.…”

“…In [4,Lemma 4.3(c)] it is shown that if (⌃, i, q) is a discrete twist, then the sets B ↵ and maps P ↵ of condition (DT2) can be chosen such that P ↵ (B ↵ \ G (0) ) ✓ ⌃ (0) . A continuous map P ↵ : B ↵ !…”

“…To finish this section, we observe that the reduction of ⌃ to any countable union of compact open subsets of G is topologically trivial. The proof of the following result is essentially the proof of [4,Theorem 4.10].…”

“…what remains should stand firmly as a substantial piece of algebra, completely accessible through algebraic avenues." The groundwork for such an extension was laid recently in [4], where a notion of an algebraic twist over an ample groupoid, suitable for constructing twisted Steinberg algebras, was developed, and the structure of the resulting twisted Steinberg algebras was analysed.…”