Locally convex algebras which determine a locally compact group
Volume 233 / 2016
Abstract
There are several algebras associated with a locally compact group $\mathcal G$ which determine ${\mathcal G}$ in the category of topological groups, such as $L^1({\mathcal G})$, $M({\mathcal G})$, 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 ${\mathcal G}_1$ and ${\mathcal G}_2$, there are infinitely many locally convex topologies $\tau _1$ and $\tau _2$ on the measure algebras $M({\mathcal G}_1)$ and $M({\mathcal G}_2)$, respectively, such that $(M({\mathcal G}_1),\tau _1)^{**}$ is isometrically isomorphic to $(M({\mathcal G}_2),\tau _2)^{**}$ if and only if ${\mathcal G}_1$ and ${\mathcal G}_2$ 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).
