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

We show that if (X.A)(X.A) and (Y,B)(Y,B) are two isomorphic Hilbert pro-C⁎C⁎-bimodules, then the crossed product A×XZA×XZ of A by X   and the crossed product B×YZB×YZ of B by Y   are isomorphic as pro-C⁎C⁎-algebras. We also prove a property of “associativity” between “⊗min⊗min” and “×X×X” as well as “⊗max⊗max” and “×X×X”.