Topological freeness for Hilbert bimodules

It is shown that topological freeness of Rieffel’s induced representation functor implies that any $C^{\ast }$-algebra generated by a faithful covariant representation of a Hilbert bimodule $X$ over a $C^{\ast }$-algebra $A$ is canonically isomorphic to the crossed product $A{⋊}_X\mathbb{Z}$. An ideal lattice description and a simplicity criterion for $A{⋊}_X\mathbb{Z}$ are established.