Crossed products by Hilbert pro-C*-bimodules versus tensor products

We show that if $(X\mathrm{.}A)$ and $(Y,B)$ are two isomorphic Hilbert pro-$C^{\ast }$-bimodules, then the crossed product $A{\times }_X\mathbb{Z}$ of $A$ by $X$ and the crossed product $B{\times }_Y\mathbb{Z}$ of $B$ by $Y$ are isomorphic as pro-$C^{\ast }$-algebras. We also prove a property of “associativity” between " ${\otimes }_{\min }$" and “${\times }_X$” \as well as " ${\otimes }_{\max }$" and “${\times }_X$”. As an application of these results we show that the crossed product of a nuclear pro-$C^{\ast }$ -algebra $A$ by a full Hilbert pro-$C^{\ast }$-bimodule $X$ is a nuclear pro-$C^{\ast }$-algebra.