The Arens-Michael envelopes of Laurent Ore extensions

TL;DR

This paper develops a topological version of Laurent tensor algebras within the framework of Arens-Michael algebras, and characterizes their envelopes, especially for invertible Ore extensions, under metrizability conditions.

Contribution

It introduces topologically invertible bimodules and constructs their Arens-Michael algebras, extending the theory of Laurent tensor algebras to a topological setting and analyzing their envelopes.

Findings

Constructed topological Laurent tensor algebras for invertible bimodules.

Provided conditions for the Arens-Michael envelope of Laurent extensions.

Proved isomorphism of envelopes for invertible Ore extensions under metrizability.

Abstract

For an Arens-Michael algebra $A$ we consider a class of $A$-$\hat{\otimes}$-bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over $A$. Given a Fr\'echet-Arens-Michael algebra $A$ and an topologically invertible Fr\'echet $A$-$\hat{\otimes}$-bimodule $M$, we construct an Arens-Michael algebra $\widehat{L}_A(M)$ which serves as a topological version of the Laurent tensor algebra $L_A(M)$. Also, for a fixed algebra $B$ we provide a condition on an invertible $B$-bimodule $N$ sufficient for the Arens-Michael envelope of $L_B(N)$ to be isomorphic to $\widehat{L}_{\widehat{B}}(\widehat{N})$. In particular, we prove that the Arens-Michael envelope of an invertible Ore extension $A[x, x^{-1}; \alpha]$ is isomorphic to $\widehat{L}_{\widehat{A}}(\widehat{A}_{\widehat{\alpha}})$ provided that the Arens-Michael envelope of $A$ is metrizable.