Convolution semigroups on hypergroups
type: Item
instance of: scholarly article
title: Convolution semigroups on hypergroups
zbMATH Open document ID: 0652.43001
author: Rupert Lasser
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: For convolution semigroups \(\mu =(\mu_ t)_{t>0}\) on abelian hypergroups K a representation theorem is derived which extends earlier results of this (``Levy-Hinčin'') type: Roughly speaking a [abelian] hypergroup is a locally compact K with involution \({\;}^-\) on K and abstract convolution * on the bounded complex Borel measures on K which is associative [commutative] with \(p_ x*p_ y(\bar M)=p_{\bar x}*p_{\bar y}(M)\) and a unit \(p_ e\), \(p_ x:=\) Dirac measure in x, supp \(p_ x*p_ y\) compact and continuous is x, y, containing e iff \(x=\bar y\). The dual \(\hat K\) contains the bounded characters, i.e. \(\omega\) : \(K\to {\mathbb{C}}\) continuous with \(\omega(x)\omega(y)=\int \omega d(p_ x*p_ y)=:p_ x*p_ y(\omega)\) for \(x,y\in K\), with the topology of uniform convergence on compacta; here it is assumed that \(\hat K\) is again a hypergroup under the canonical product. A \(\mu: (0,\infty)\to\) nonnegative bounded Borel measures on K is called a convolution semigroup if \(\mu_ t(K)\leq 1\), \(\mu_ s*\mu_ t=\mu_{s+t}\) for \(s,t>0\) and \(\mu_ t\to p_ e\) if \(t\downarrow 0\). To such a \(\mu\) there exists a unique negative definite \(\phi: K\to {\mathbb{C}}\) with Re\(\phi\geq 0\) and \({\hat\mu}{}_ t(\omega)=\exp (- t\phi(\omega))\), \(\omega\in \hat K\), \(t>0\), the ``associated'' \(\phi\) ; \(\phi\) negative definite means \[ \sum_{j,k}(\phi (x_ j)+\overline{\phi(x_ k)}-p_{x_ j}*p_{x_ k}(\phi))c_ j\overline{c_ k}\geq 0 \] for all \(x_ 1,...,x_ n\in K\), \(c_ j\in {\mathbb{C}}\); examples of negative definite functions are \(\phi=ih\), \(h: K\to {\mathbb{R}}\) with \(p_ x*p_ y(h)=h(x)+h(y)\) and \(h(\bar x)=-h(x)\), or quadratic forms \(q>0\) (i.e. q: \(K\to {\mathbb{R}}\) with \(p_ x*p_{\bar y}(q)+p_ x*p_ y(q)=2(q(x)+q(y)))\). Under an additional assumption (F) on compact sets of \(\hat K\) the following Levy-Hinčin representation formula for convolution semigroups with symmetric Levy measure \(\lambda:=\lim_{t\downarrow 0}(1/t)\mu_ t\) is obtained, …
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 43A35
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 22B99
Mathematics Subject Classification ID: 22A30
zbMATH DE Number: 4063760
zbMATH Keywords: convolution semigroups
zbMATH Keywords: abelian hypergroups
zbMATH Keywords: representation theorem
zbMATH Keywords: bounded complex Borel measures
zbMATH Keywords: bounded characters
zbMATH Keywords: negative definite functions
zbMATH Keywords: quadratic forms
zbMATH Keywords: Levy-Hinčin representation
zbMATH Keywords: symmetric Levy measure
zbMATH Keywords: Locally compact abelian groups
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Gelfand pairs
zbMATH Keywords: Bessel functions
reviewed by: Hans Günzler
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.353
OpenAlex ID: W1591125299
DOI: 10.2140/PJM.1987.127.353
description: scientific article; zbMATH DE number 4063760