A new approach to convolution and semi-direct products of groups

Abstract: Let $H$ and $K$ be locally compact groups and $\tau:H\to Aut(K)$ be a continuous homomorphism and also let $G_\tau=H\ltimes_\tau K$ be the semi-direct product of $H$ and $K$ with respect to $\tau$. We define left and also right $\tau$-convolution on $L^1(G_\tau)$ such that $L^1(G_\tau)$ with respect to each of them is a Banach algebra. Also we define $\tau$-convolution as a linear combination of the left and right $\tau$-convolution. We show that the $\tau$-convolution is commutative if and only if $K$ is abelian and also when $H$ and $K$ are second countable groups, the $\tau$-convolution coincides with the standard convolution of $L^1(G_\tau)$ if and only if $H$ is the trivial group. We prove that there is a $\tau$-involution on $L^1(G_\tau)$ such that $L^1(G_\tau)$ with respect to the $\tau$-involution and $\tau$-convolution is a non-associative Banach *-algebra and also it is also shown that when $K$ is abelian, the $\tau$-involution and $\tau$-convolution makes $L^1(G_\tau)$ into a Jordan Banach *-algebra.