The imprimitivity Fell bundle

Abstract Given a full right-Hilbert $\mathrm {C}^{*}$ -module $\mathbf {X}$ over a $\mathrm {C}^{*}$ -algebra A , the set $\mathbb {K}_{A}(\mathbf {X})$ of A -compact operators on $\mathbf {X}$ is the (up to isomorphism) unique $\mathrm {C}^{*}$ -algebra that is strongly Morita equivalent to the coefficient algebra A via $\mathbf {X}$ . As a bimodule, $\mathbb {K}_{A}(\mathbf {X})$ can also be thought of as the balanced tensor product $\mathbf {X}\otimes _{A} \mathbf {X}^{\mathrm {op}}$ , and so the latter naturally becomes a $\mathrm {C}^{*}$ -algebra. We generalize both of these facts to the world of Fell bundles over groupoids: Suppose $\mathscr {B}$ is a Fell bundle over a groupoid $\mathcal {H}$ and $\mathscr {M}$ is an upper semi-continuous Banach bundle over a principal $\mathcal {H}$ -space X . If $\mathscr {M}$ carries a right-action of $\mathscr {B}$ and a sufficiently nice $\mathscr {B}$ -valued inner product, then its imprimitivity Fell bundle $\mathbb {K}_{\mathscr {B}}(\mathscr {M})=\mathscr {M}\otimes _{\mathscr {B}} \mathscr {M}^{\mathrm {op}}$ is a Fell bundle over the imprimitivity groupoid of X , and it is the unique Fell bundle that is equivalent to $\mathscr {B}$ via $\mathscr {M}$ . We show that $\mathbb {K}_{\mathscr {B}}(\mathscr {M})$ generalizes the “higher order” compact operators of Abadie–Ferraro in the case of saturated bundles over groups, and that the theorem recovers results such as Kumjian’s Stabilization trick.