Let $X$ be a hypergroup. In this paper, we define a locally convex topology $\beta $ on $L(X)$ such that $(L(X),\beta )^*$ with the strong topology can be identified with a Banach subspace of $L(X)^*$. We prove that if $X$ has a Haar measure, then the dual to this subspace is $L_C(X)^{**}= \operatorname{cl}\{F\in L(X)^{**}; F$ has compact carrier\}. Moreover, we study the operators on $L(X)^*$ and $L_0^\infty(X)$ which commute with translations and convolutions. We prove, among other things, that if $\operatorname{wap}(L(X))$ is left stationary, then there is a weakly compact operator $T$ on $L(X)^*$ which commutes with convolutions if and only if $L(X)^{**}$ has a topologically left invariant functional. For the most part, $X$ is a hypergroup not necessarily with an involution and Haar measure except when explicitly stated.
@article{119421,
author = {Ali Ghaffari},
title = {Convolution operators on the dual of hypergroup algebras},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {44},
year = {2003},
pages = {669-679},
zbl = {1098.43001},
mrnumber = {2062883},
language = {en},
url = {http://dml.mathdoc.fr/item/119421}
}
Ghaffari, Ali. Convolution operators on the dual of hypergroup algebras. Commentationes Mathematicae Universitatis Carolinae, Tome 44 (2003) pp. 669-679. http://gdmltest.u-ga.fr/item/119421/
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