Involutions in algebras related to second duals of hypergroup algebras

Abstract

Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K{)}^{\ast \ast }$, $L(K{)}^{\ast \ast }$, and $L_c(K{)}^{\ast \ast }$. We show that the natural involution of $M\left(K\right)$ has the canonical extension to $M(K{)}^{\ast \ast }$ if and only if the natural involution of $L\left(K\right)$ has the canonical extension to $L(K{)}^{\ast \ast }$. Also, we give necessary and sufficient conditions for $M(K{)}^{\ast \ast }$ and $L(K{)}^{\ast \ast }$ to admit an involution extending the natural involution of $M\left(K\right)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_c(K{)}^{\ast \ast }$ to admit an involution.