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

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

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