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