The Arens-Michael envelopes of Laurent Ore extensions

For an Arens-Michael algebra $A$ we consider a class of $A$-$\widehat{\otimes }$-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 $A$-$\widehat{\otimes }$-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 $A[x, x^{-1}; \alpha]$isisomorphicto$\widehat{L}{\widehat{A}}(\widehat{A}{\widehat{\alpha}})$ provided that the Arens-Michael envelope of $A$ is metrizable.