Cross-sectional C*-algebras of Fell bundles

Abstract: A Fell bundle (or C*-algebraic bundle) $B=\{B_t\}_{t\in G}$ may be thought of as a kind of action of the base group $G$ on the C*-algebra $B_e$, $e$ being the unit of $G.$ When doing so, the full and reduced cross-sectional C*-algebras of $B$, $C^*(B)$ and $C^*_r(B)$ respectively, become the full and reduced crossed products of the action.

It is implicit in Exel-Ng's construction/characterization of $C^*_r(B)$ that the induction of representations from $H:=\{e\}$ to $G,$ $U\mapsto Ind_{H\uparrow G}(U),$ and from $B_e$ to $B,$ $T\mapsto Ind_{B_e\uparrow B}(T),$ are intimately related and that both can be used to define/describe $C^*_r(B).$ If $B$ is saturated, the equivalence of the definitions is a straightforward consequence of Fell's absorption principle.

The situation is not so clear when one considers closed subgroups $H$ of $G$ other than $\{e\}$ (even if $B$ is saturated). The reduction of $B$ to $H,$ $B_H:=\{B_t\}_{t\in H},$ is a Fell bundle and one has induction processes $U\mapsto Ind_{H\uparrow G}(U)$ and $T\mapsto Ind_{B_H\uparrow B}(T),$ where $U$ and $T$ stand for representations of $H$ and $B_H,$ respectively. In this talk we use $U\mapsto Ind_{H\uparrow G}(U)$ and $T\mapsto Ind_{B_H\uparrow B}(T)$ to construct two candidates for the "reduced $H$-cross-sectional C*-algebra of $B$". We also give conditions implying they are isomorphic.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( paper | video )

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