Non-archimedean harmonic analysis on topological semigroups

TLDR: In this paper, the Fourier-Stieltjes transform is isometric for topological semi-lattices and the convolutional Banach algebra of all tight K-valued measures on S and its ideal Ma(S) is studied.

Abstract: Let K be a complete non-Archimedean valued field, S a commutative topological semigroup (not necessarily locally compact). We study the convolution Banach algebra M(S) of all tight K-valued measures on S and its ideal Ma(S) consisting of all elements μ of M(S) for which the shift map x ↦ x ∗μ is continuous (the analogue of the group algebra). In particular, we show that under reasonable conditions the Fourier-Stieltjes transform is isometric. Furthermore, we describe the images of M(S) and Ma(S) under the Fourier-Stieltjes transform. The results obtained are known for groups but new for topological (semi) lattices.