ABSTRACT STRUCTURE OF PARTIAL FUNCTION -ALGEBRAS OVER SEMI-DIRECT PRODUCT OF LOCALLY COMPACT GROUPS

This article presents a uni ed 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 Kbe locally compact groups and : H ! Aut(K) be a continuous homomorphism. Let G = H ⋉ K be the semi-direct product ofH and K with respect to . We de ne left and right -convolution on L1(G ) and we show that, with respect to each of them, thefunction space L1(G ) is a Banach algebra. We de ne -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 anon-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.