Multipliers of Hilbert pro-C*-bimodules and crossed products by Hilbert pro-C*-bimodules

In this paper we introduce the notion of multiplier of a Hilbert pro-$C^{\ast }$-bimodule and we investigate the structure of the multiplier bimodule of a Hilbert pro-$C^{\ast }$-bimodule. We also investigate the relationship between the crossed product $A{\times }_X\mathbb{Z}$ of a pro-$$-algebra $A$ by a Hilbert pro-$C^{\ast }$-bimodule $X$ over $A$, the crossed product $M(A){\times }_{M(X)}\mathbb{Z}$ of the multiplier algebra $M(A)$ of $A$ by the multiplier bimodule $M(X)$ of $X$ and the multiplier algebra $M(A\times _{X}$\mathbb{Z}$)$of$A\times _{X}$\mathbb{Z}$$.