The Imprimitivity Fell Bundle

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