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Search: MSC category 43A62 ( Hypergroups )

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1. CMB 2017 (vol 60 pp. 402)

Shravan Kumar, N.
Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II
Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let $VN(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, we show that the set of invariant means on $VN(K)$ is singleton if and only if $K$ is discrete. Here $K$ need not be second countable. We also study invariant means on the dual of the Fourier algebra $A_0(K),$ the closure of $A(K)$ in the $cb$-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

Keywords:ultraspherical hypergroup, Fourier algebra, Fourier-Stieltjes algebra, invariant mean, generalized translation, generalized invariant mean
Categories:43A62, 46J10, 43A30, 20N20

2. CMB 2015 (vol 58 pp. 415)

Willson, Benjamin
A Fixed Point Theorem and the Existence of a Haar Measure for Hypergroups Satisfying Conditions Related to Amenability
In this paper we present a fixed point property for amenable hypergroups which is analogous to Rickert's fixed point theorem for semigroups. It equates the existence of a left invariant mean on the space of weakly right uniformly continuous functions to the existence of a fixed point for any action of the hypergroup. Using this fixed point property, a certain class of hypergroups are shown to have a left Haar measure.

Keywords:invariant measure, Haar measure, hypergroup, amenability, function translations
Categories:43A62, 43A05, 43A07

3. CMB 2010 (vol 53 pp. 491)

Jizheng, Huang; Liu, Heping
The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy--Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.

Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel
Categories:42B25, 43A62

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