Some consequences of spectral synthesis in hypergroup algebras

Abstract

Properties of spectral synthesis are exploited to show that, for a large class of commutative hypergroups and for every compact hypergroup, every closed, reflexive, left-translation-invariant subspace of $L^{\infty }\left(K\right)$ is finite-dimensional. Also, we show that, for a class of hypergroups which includes many commutative hypergroups and all $\mathcal{Z}$-hypergroups, every derivation of $L^1\left(K\right)$ into an arbitrary Banach $L^1$-bimodule is continuous.