Bochner's Theorem and the Hausdorff Moment Theorem on foundation Topological Semigroups

Abstract

One of the most basic theorems in harmonic analysis on locally compact commutative groups is Bochner's theorem (see [16, p. 19]). This theorem characterizes the positive definite functions. In 1971, R. Lindhal and P. H. Maserick proved a version of Bochner's theorem for discrete commutative semigroups with identity and with an involution * (see [13]). Later, in 1980, C. Berg and P. H. Maserick in [6] generalized this theorem for exponentially bounded positive definite functions on discrete commutative semigroups with identity and with an involution *. In this work we develop these results, and also the Hausdorff moment theorem, for an extensive class of topological semigroups, the so-called “foundation topological semigroups”. We shall give examples to show that these theorems do not extend in the obvious way to general locally compact topological semigroups.

A generalization of Bochner-Weil?s theorem and Stone?s theorem on foundation semigroups

“…It also generalizes our previous version of the Bochner theorem (Theorem 2.12 of [6]), via quite a different technique of proof, from the case of Borel measurable weights w for which both w and 1 w are locally bounded to the case of an arbitrary weight. …”

Positive definite and related functions on semigroups

“…On the other hand the continuous positive definite functions on [0, 1] are the non-negative, continuous and decreasing functions, so a topological version of Theorem 1.1 is not true without some restrictions. A paper by LASHKARIZADEH BAMI [28] contains a version of Theorem 1 .3 for α-bounded continuous positive definite functions on foundation topological semigroups.…”

“…One simply has to define σ(Α) = σ Ε (π Ε (Α)) for A e A(S*), (28) where E G T>o(S) is chosen such that A Ε π Ε \Β(Ε*)). The right-hand side of (28) turns out to be independent of the choice of E. The stability property (i) depends on a result about bimeasures like the corresponding result for Radon perfectness.…”

Restricted algebras on inverse semigroups I, representation theory

“…If G is a locally compact Abelian group, A(G) and B(G) are the ranges of the Fourier and Fourier-Stieltjes transforms on L 1 (G) and M (G), respectively. Dunkl and Ramirez defined a subalgebra R(S) of the algebra WAP(S) for a Hausdorff locally compact commutative topological semigroup S in [9](see also [13],14]). For a locally compact Abelian group G, R(G) = M ( Ĝ) ˆ.…”

“…In this paper, our aim is to study the restricted Fourier and Fourier-Stieltjes algebras A(S) and B(S) on an inverse semigroup S. In particular, we prove restricted version of the Eymard's characterization [5] of the Fourier algebra (Theorem 2.1).The structure of algebras B(S) and A(S) is far from being well understood, even in special cases. From the results of [4], [7], it is known that for a commutative unital discrete * -semigroup S, B(S) = M ( Ŝ) ˆvia Bochner theorem [7]. Even in this case, the structure of A(S) seems to be much more complicated than the group case.…”