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

TL;DR

This paper demonstrates that isomorphic Hilbert pro-C*-bimodules lead to isomorphic crossed products and explores the interaction between tensor products and crossed products, with applications to nuclear pro-C*-algebras.

Contribution

It establishes the isomorphism of crossed products for isomorphic bimodules and investigates their algebraic properties, including a form of associativity with tensor products.

Findings

Crossed products are isomorphic for isomorphic bimodules.

Identifies a property of associativity between tensor and crossed products.

Shows nuclearity is preserved under crossed products with full bimodules.

Abstract

We show that if $(X.A)$ 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.