The Arens-Michael envelopes of Laurent Ore extensions

Abstract

For an Arens-Michael algebra A we consider a class of A-⊗ˆ -bimodules which are invertible with respect to the projective bimodule tensor product. We call such bimodules topologically invertible over A. Given a Fréchet-ArensMichael algebra A and a topologically invertible Fréchet A-⊗ˆ -bimodule M , we construct an Arens-Michael algebra LbA(M) which serves as a topological version of the Laurent tensor algebra LA(M). Also, for a fixed algebra B we provide a condition on an invertible B -bimodule N which allows us to explicitly describe the Arens-Michael envelope of LB(N) as a topological Laurent tensor algebra. In particular, we provide an explicit description of the Arens-Michael envelope of an invertible Ore extension $A[x, x^{−1}; α]$for a metrizable algebra A.