Locally convex algebras which determine a locally compact group
There are several algebras associated with a locally compact group 𝓖 which determine 𝓖 in the category of topological groups, such as L¹(𝓖), M(𝓖), and their second duals. In this article we add a fairly large family of locally convex algebras to this list. More precisely, we show that for two infinite locally compact groups 𝓖₁ and 𝓖₂, there are infinitely many locally convex topologies τ₁ and τ₂ on the measure algebras M(𝓖₁) and M(𝓖₂), respectively, such that (M(𝓖₁),τ₁)** is isometrically isomorphic to (M(𝓖₂),τ₂)** if and only if 𝓖₁ and 𝓖₂ are topologically isomorphic. In particular, this leads to a new proof of Ghahramani-Lau's isometrical isomorphism theorem for compact groups, different from those of Ghahramani and J. P. McClure (2006) and Dales et al. (2012).
- Department of Mathematics, Faculty of Sciences, Shiraz University, Shiraz 71454, Iran
- Department of Mathematics, Yazd University, P.O. Box 89195-741, Yazd, Iran
- Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran
- School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran