## Induced representation theories between equivalent Fell bundles

### Volume 273 / 2023

#### Abstract

We construct a tool consisting of three theorems revealing the connection of the induced representation theories between two equivalent Fell bundles, with the aid of which we may transfer induced representation-theoretic theorems for saturated Fell bundles to arbitrary Fell bundles. Letting $G$ be a locally compact group with closed subgroup $H$ and $\mathscr {B}$ a Fell bundle over $G$, as applications of our main result we prove that any $\ast $-representation of the restricted bundle $\mathscr {B}_H$ can be induced to $\mathscr {B}$; and we show that $C^{\ast }(\mathscr {B}_H)$ and $C^{\ast }(\mathscr {D})$, where $\mathscr {D}$ is the $G, G/H$ transformation bundle derived from $\mathscr {B}$, are Morita equivalent if and only if $C^{\ast }(\mathscr {B}_H)$ and $C^{\ast }(\mathscr {B}_{xHx^{-1}})$ are Morita equivalent for any $x \in G$.