in this article, by using a partial on locally compact semi-direct product groups, we present a compatible extension of the fourier transform. as a consequence, we extend the fundamental theorems of abelian fourier transform to non-abelian case.

A locally compact group G is amenable if and only if it has Reiter’s property (Pp) for p = 1 or, equivalently, all p ∈ [1,∞), i.e., there is a net (mα)α of non-negative norm one functions in L(G) such that limα supx∈K ‖Lx−1mα − mα‖p = 0 for each compact subset K ⊂ G (Lx−1mα stands for the left translate of mα by x ). We extend the definitions of properties (P1) and (P2) from locally compact gro...