Induction, absorption and weak containment of *-representations of Banach *-algebraic bundles

TL;DR

This paper investigates the induction and weak containment of *-representations of Banach *-algebraic bundles over locally compact groups, establishing conditions for faithfulness of associated quotient maps and constructing related Fell bundles.

Contribution

It introduces a framework for inducing *-representations via Fell's process, analyzes conditions for faithfulness of quotient maps, and constructs new Fell bundles linking different subgroup structures.

Findings

Induction of *-representations extends from subgroups to the whole group.

Faithfulness of quotient maps depends on properties like saturation and nuclearity.

Construction of Fell bundles over quotient groups relates different crossed product C*-algebras.

Abstract

Given a Fell bundle $\mathcal{B}=\{B_t\}_{t\in G}$ over a LCH group and a closed subgroup $H\subset G,$ we show that all the *-representations of $\mathcal{B}_H:=\{B_t\}_{t\in H}$ can be induced to *-representations of $\mathcal{B}$ by means of Fell's induction process; which we describe as induction via a *-homomorphism $q^{\mathcal{B}}_H\colon C^*(\mathcal{B})\to \mathbb{B}(X_{C^*(\mathcal{B}_H)}).$ The quotients $C^*_H(\mathcal{B}):=q^{\mathcal{B}}_H(C^*(\mathcal{B}))$ are intermediate to $C^*(\mathcal{B})= C^*_G(\mathcal{B})$ and $C^*_{r}(\mathcal{B})=C^*_{\{e\}}(\mathcal{B})$ because every inclusion of subgroups $H\subset K\subset G$ gives a unique quotient map $q^{\mathcal{B}}_{HK}\colon C^*_K(\mathcal{B})\to C^*_H(\mathcal{B})$ such that $q^{\mathcal{B}}_{HK}\circ q^{\mathcal{B}}_K=q^{\mathcal{B}}_H.$ All along the article we try to find conditions on $\mathcal{B},$ $G,\ H$ and $K$ (e.g. saturation, nuclearity or weak containment) that imply $q^{\mathcal{B}}_{HK}$ is faithful. One of our main tools is a blend of Fell's absorption principle (for saturated bundles) and a result of Exel and Ng for reduced cross sectional C*-algebras. We also show that given an imprimitivity system $\langle T,P\rangle$ for $\mathcal{B}$ over $G/H,$ if $H$ is open or has open normalizer in $G,$ then $T$ is weakly contained in a *-representation induced from $\mathcal{B}_H$ (even if $\mathcal{B}$ is not saturated). Given normal and closed subgroups of $G,$ $H\subset K,$ we construct a Fell bundle $\mathcal{C}$ over $G/K$ such that $C^*_r(\mathcal{C})=C^*_H(\mathcal{B}).$ We show that $q^{\mathcal{B}}_H$ is faithful if and only if both $q^{\mathcal{C}}_{\{e\}}$ and $q^{\mathcal{B}_K}_H$ are.