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April 2022 On the structure of hypergroups with respect to the induced topology
Ali Ghaffari, Tahere Haddadi, Samaneh Javadi, Marjan Sheibani Abdolyousefi
Rocky Mountain J. Math. 52(2): 519-533 (April 2022). DOI: 10.1216/rmj.2022.52.519

Abstract

Let H be a hypergroup with left Haar measure and let L1(H) be the complex Lebesgue space associated with it. Let L(H) be the set of all locally measurable functions that are bounded except on a locally null set, modulo functions that are zero locally a.e. It is a standard device to embed L(H) into (L1(H),L(H)). We denote the strong operator topology and the weak operator topology on L(H) by τso and τwo. Unlike the uniform norm and weak topologies on L(H), they depend essentially on the hypergroup structure of H. We derive that the τso-topology is always different from the weak-topology whenever H is infinite. We can conclude that for a compact hypergroup H, L1(H) is the dual of (L(H),τso). The properties of τso and τwo are then studied further and we pay attention to the τwo-almost periodic elements of L(H). Finally we give some further results about bounded linear operators which are τso-τso-continuous.

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Ali Ghaffari. Tahere Haddadi. Samaneh Javadi. Marjan Sheibani Abdolyousefi. "On the structure of hypergroups with respect to the induced topology." Rocky Mountain J. Math. 52 (2) 519 - 533, April 2022. https://doi.org/10.1216/rmj.2022.52.519

Information

Received: 8 November 2020; Revised: 18 June 2021; Accepted: 9 July 2021; Published: April 2022
First available in Project Euclid: 17 May 2022

Digital Object Identifier: 10.1216/rmj.2022.52.519

Subjects:
Primary: 43A15
Secondary: 43A22 , 43A62

Keywords: amenability , Banach algebras , hypergroup algebras , strong operator topology , weak operator topology , weak∗-continuous , weakly almost periodic functions

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 2 • April 2022
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