Abstract structure of partial function $*$-algebras over semi-direct product of locally compact groups

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 $tau:Hto Aut(K)$ be a continuous homomorphism.  Let $G_tau=Hltimes_tau K$ 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 $p>1$.