An equivalence theorem for reduced Fell bundle \(C^\ast\)-algebras (Q374057)

The authors prove that equivalent Fell bundles over groupoids have Morita equivalent reduced cross-sectional \(C^*\)-algebras. The main technical innovation is the introduction of a linking bundle whose cross-sectional algebra contains information about the Morita equivalence between the full and the reduced algebras of the given bundles. More precisely, let \(\mathcal{B}\) and \(\mathcal{C}\) be equivalent upper semicontinuous Fell bundles over groupoids \(G\) and \(H\), respectively; see [\textit{P. S. Muhly} and \textit{D. P. Williams}, Diss. Math. 456 (2008; Zbl 1167.46040)]. The existence of an equivalence includes existence of an upper semicontinuous Banach bundle \(\mathcal{E}\) over a space \(Z\) such that \(G\) and \(H\) are equivalent groupoids via \(Z\), in Renault's sense.NEWLINENEWLINEThe present authors construct a linking bundle \({L}(\mathcal{E})=\mathcal{B} \sqcup \mathcal{E}\sqcup \mathcal{E}^{\text{op}}\sqcup\mathcal{C}\) whose associated completion in the full and the reduced norm, respectively, provides a linking algebra for a Morita equivalence between on the one hand \(C^*(G,\mathcal{B})\) and \(C^*(H,\mathcal{C})\) and on the other \(C_r^*(G,\mathcal{B})\) and \(C_r^*(H,\mathcal{C})\). The Morita equivalence of the full \(C^*\)-algebras had been obtained earlier by Muhly and Williams [loc. cit.], and of the reduced \(C^*\)algebras in the \( r\)-discrete case by \textit{A. Kumjian} [Proc. Am. Math. Soc. 126, No. 4, 1115--1125 (1998; Zbl 0891.46038)]. Applications and relations to earlier results generalised in the present work are considered.