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.