%0 Journal Article
%T Abstract structure of partial function $*$-algebras over semi-direct product of locally compact groups
%J Sahand Communications in Mathematical Analysis
%I University of Maragheh
%Z 2322-5807
%A Ghaani Farashahi, Arash
%A Kamyabi-Gol, Rajab Ali
%D 2015
%\ 12/01/2015
%V 02
%N 2
%P 23-44
%! Abstract structure of partial function $*$-algebras over semi-direct product of locally compact groups
%K Semi-direct products of groups
%K Left $tau$-convolution ($tau_l$-convolution)
%K Right $tau$-convolution
($tau_r$-convolution)
%K $tau$-convolution
%K $tau$-involution
%K $tau$-approximate identity
%R
%X This article presents a unified approach to the abstract notions of partial convolution and involution in $L^p$-function spaces over semi-direct product of locally compact groups. Let $H$ and $K$ be locally compact groups and $\tau:H\to Aut(K)$ be a continuous homomorphism. Let $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to $\tau$. We define left and right $\tau$-convolution on $L^1(G_\tau)$ and we show that, with respect to each of them, the function space $L^1(G_\tau)$ is a Banach algebra. We define $\tau$-convolution as a linear combination of the left and right $\tau$-convolution and we show that the $\tau$-convolution is commutative if and only if $K$ is abelian. We prove that there is a $\tau$-involution on $L^1(G_\tau)$ such that with respect to the $\tau$-involution and $\tau$-convolution, $L^1(G_\tau)$ is a non-associative Banach $*$-algebra. It is also shown that when $K$ is abelian, the $\tau$-involution and $\tau$-convolution make $L^1(G_\tau)$ into a Jordan Banach $*$-algebra. Finally, we also present the generalized notation of $\tau$-convolution for other $L^p$-spaces with $p>1$.
%U https://scma.maragheh.ac.ir/article_15512_5770f8eeb189b81deec09dc87fdd4b39.pdf