The Bochner-Schoenberg-Eberlein Property for Totally Ordered Semigroup Algebras.

The concepts of BSE property and BSE algebras were introduced and studied by Takahasi and Hatori in 1990 and later by Kaniuth and Ülger. This abbreviation refers to a famous theorem proved by Bochner and Schoenberg for $$L^1({\mathbb {R}})$$ , where $${\mathbb {R}}$$ is the additive group of real numbers, and by Eberlein for $$L^1(G)$$ of a locally compact abelian group G. In this paper we investigate this property for the Banach algebra $$L^p(S,\mu )\;(1\le p<\infty )$$ where S is a compact totally ordered semigroup with multiplication $$xy=\max \{x,y\}$$ and $$\mu $$ is a regular bounded continuous measure on S. As an application, we have shown that $$L^1(S,\mu )$$ is not an ideal in its second dual. [ABSTRACT FROM AUTHOR]