On Imprimitivity Hilbert Bimodules Over Commutative $H^*$-Algebras

M. Khanehgir, M. Moradian Khibary, Z. Niazi Moghani

In this paper, we introduce the notion of imprimitivity Hilbert $H^*$-bimodule and describe some properties of it. Moreover, we show that if $\mathcal{A}$ and $\mathcal{B}$ are proper and commutative $H^*$-algebras, $_\mathcal{A}E_\mathcal{B}$ is a Hilbert $H^*$-bimodule and $e_1$ is a minimal projection in $\mathcal{A}$ with $_\mathcal{A}[x|x]=e_1$ for some $x\in \mathcal{A}$, then $[x|x]_\mathcal{B}$ is a minimal projection in $\mathcal{B}$, too. Furthermore, the existence of orthonormal bases for such spaces is studied.