BSE norms and BSE algebras

Abstract

Let G be a locally compact abelian group, with dual group $\Gamma$. The classical Bochner–Schoenberg–Eberlein theorem states the following. Take $f\in C^{\,b}(\Gamma )$. Then $f = \widehat{\mu }$ for some $\mu \in M(G)$ if and only if there is a constant $\beta \ge 0$ with the following property: for each $n\in \mathbb N$, each $\gamma _1,\dots , \gamma _n \in \Gamma$, and each $\alpha _1,\dots \alpha _n\in \mathbb C$, necessarily $\left| \sum _{i=1}^n\alpha _if(\gamma _i)\right| \le \beta \left\| \sum _{i=1}^n \alpha _i \gamma _i\right\| _{L^{\infty }(G)}\,.$ Further, in this case, the infimum of the constants $\beta$ that satisfy the above inequality is $\left\| \mu \right\|$. This theorem is proved in the text of Rudin [276, Theorem 1.9.1], for example. This basic theorem for abelian groups was proved by Bochner [22] in the case where $\Gamma =\mathbb R$; an integral analogue was given by Schoenberg [291]; the case for general abelian groups was given by Eberlein in [92].