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 $p>1$.