Configurations and invariant nets for amenable hypergroups and related algebras

Willson, Benjamin

doi:10.1090/S0002-9947-2014-05731-0

Configurations and invariant nets for amenable hypergroups and related algebras
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Trans. Amer. Math. Soc. 366 (2014), 5087-5112 Request permission

Abstract:

Let $H$ be a hypergroup with left Haar measure. The amenability of $H$ can be characterized by the existence of nets of positive, norm one functions in $L^1(H)$ which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of $H$ using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in $L^1(H)$ which tend to left invariance weakly, but not in norm.

We define the semidirect product of $H$ with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product.

These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra.

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