The Arens-Michael envelopes of Laurent Ore extensions

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 Fréchet-Arens-Michael algebra $A$ and an topologically invertible Fréchet $A$-$\hat{\otimes }$-bimodule $M$, we construct an Arens-Michael algebra ${\hat{L}}_A(M)$ 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 ${\hat{L}}_{\hat{B}}(\hat{N})$. In particular, we prove that the Arens-Michael envelope of an invertible Ore extension $A[x,x^{-1};\alpha ]$ is isomorphic to ${\hat{L}}_{\hat{A}}({\hat{A}}_{\hat{\alpha }})$ provided that the Arens-Michael envelope of $A$ is metrizable.

arXiv comment22 pages, the author gave a talk based on the results of the paper, at the conference "Banach Algebras and Applications", 3-11 July, Oulu