Locally convex algebras which determine a locally compact group

Gholam Hossein Esslamzadeh ¹   ; Hossein Javanshiri ²   ; Rasoul Nasr-Isfahani ³

¹ Department of Mathematics Faculty of Sciences Shiraz University Shiraz 71454, Iran
² Department of Mathematics Yazd University P.O. Box 89195-741 Yazd, Iran
³ Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156-83111, Iran and School of Mathematics Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5746 Tehran, Iran

Studia Mathematica, Tome 233 (2016) no. 3, pp. 197-207

Résumé

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).

Détail

DOI : 10.4064/sm7879-4-2016

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