Classification of One-Dimensional Hypergroups

Abstract

$(H, * )$ is a hypergroup if $H$ is a locally compact space and * is a binary operation with respect to which $M(H)$, the bounded Borel measures on $H$, becomes a Banach algebra with a number of additional properties so that * generalizes group convolution. The case when $H$ is a one-dimensional set, that is, a circle or an interval (possibly unbounded), includes a large number of examples: the classical group algebras $M(R)$ and $M(T)$ ($T$ is the unit circle), the subalgebra of $M(R)$ consisting of even measures, the subalgebra of $M({R^n})$ consisting of rotation invariant measures, and the subalgebra of $M({S^n})$ (${S^n}$ is the unit sphere in ${R^{n + 1}}$) consisting of zonal measures. In addition to these, there are several continua of measure algebras unrelated to groups or geometry such as the measure algebras which arise in connection with Hankel transforms, ultraspherical and Jacobi polynomial series, and Sturm-Liouville expansions. The main result of this article is a classification of the one-dimensional hypergroups. It is shown that if a certain amount of "regularity" for * is assumed, then every one-dimensional hypergroup is commutative and, up to a change of variables, must be one of the following types: (i) the classical group algebra $M(R)$, (ii) the classical group algebra $M(T)$, (iii) $(H{,^*})$, where $H = [0,P]$ or $[0,\infty )$, ${\delta _0}$ is an identity for *, and $0 \in \operatorname {supp}({\delta _x}*{\delta _y})$ if and only if $x = y$.