Wavelet and Linear Algebra

Abstract In this paper we develop the notions of Connes amenability for certain product of Banach algebras. We give necessary and sufficient conditions for the existence of an invariant mean on the predual of $\Theta$-Lau product  $\mathcal{A}\times_{\Theta}\mathcal{B}$, module extension Banach algebra $\mathcal{A}\oplus\mathcal{X}$ and projective tensor product $\mathcal{A} \widehat{{\otimes}} \mathcal{B}$, where $\mathcal{A}$ and $\mathcal{B}$ are dual Banach algebras  with preduals $\mathcal{A}_*$ and $\mathcal{B}_*$ respectively and $\mathcal{X}$ is a normal Banach $\mathcal{A}$-bimodule with predual $\mathcal{X}_*$.

Abstract Let $K$ be a locally compact hypergroup with left Haar measure and let $L^1(K)$ be the complex Lebesgue space associated with it. Let $L^\infty(K)$ be the dual of $L^1(K)$. The purpose of this paper is to present some necessary and sufficient conditions for $L^\infty(K)^*$ to have a topologically left invariant mean. Some
characterizations of amenable hypergroups are given.