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 Lp-function spaces over semi-direct product of locally compact groups. Let H and K be locally compact groups and τ:H→Aut(K) be a continuous homomorphism. Let Gτ=H?τK be the semi-direct product of H and K with respect to τ. We define left and right τ-convolution on L1(Gτ) and we show that, with respect to each of them, the function space L1(Gτ) is a Banach algebra. We define τ-convolution as a linear combination of the left and right τ-convolution and we show that the τ-convolution is commutative if and only if K is abelian. We prove that there is a τ-involution on L1(Gτ) such that with respect to the τ-involution and τ-convolution, L1(Gτ) is a non-associative Banach ?-algebra. It is also shown that when K is abelian, the τ-involution and τ-convolution make L1(Gτ) into a Jordan Banach ?-algebra. Finally, we also present the generalized notation of τ-convolution for other Lp-spaces with p>1.