Convolution operators on the dual of hypergroup algebras

Ali Ghaffari

Commentationes Mathematicae Universitatis Carolinae (2003)

  • Volume: 44, Issue: 4, page 669-679
  • ISSN: 0010-2628

Abstract

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Let X be a hypergroup. In this paper, we define a locally convex topology β on L ( X ) such that ( L ( X ) , β ) * 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 ) * * = cl { F ∈ L ( X ) * * ; F has compact carrier}. Moreover, we study the operators on L ( X ) * and L 0 ∞ ( X ) which commute with translations and convolutions. We prove, among other things, that if 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.

How to cite

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Ghaffari, Ali. "Convolution operators on the dual of hypergroup algebras." Commentationes Mathematicae Universitatis Carolinae 44.4 (2003): 669-679. <http://eudml.org/doc/249195>.

@article{Ghaffari2003,
abstract = {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 $L_C(X)^\{**\}= \operatorname\{cl\}\lbrace F\in L(X)^\{**\}; F$ 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.},
author = {Ghaffari, Ali},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Arens regular; hypergroup algebra; weakly almost periodic; convolution operators; Arens regular; hypergroup algebra; weakly almost periodic; convolution operators},
language = {eng},
number = {4},
pages = {669-679},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Convolution operators on the dual of hypergroup algebras},
url = {http://eudml.org/doc/249195},
volume = {44},
year = {2003},
}

TY - JOUR
AU - Ghaffari, Ali
TI - Convolution operators on the dual of hypergroup algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2003
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 44
IS - 4
SP - 669
EP - 679
AB - 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 $L_C(X)^{**}= \operatorname{cl}\lbrace F\in L(X)^{**}; F$ 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.
LA - eng
KW - Arens regular; hypergroup algebra; weakly almost periodic; convolution operators; Arens regular; hypergroup algebra; weakly almost periodic; convolution operators
UR - http://eudml.org/doc/249195
ER -

References

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