BSE norms and BSE algebras

Let G be a locally compact abelian group, with dual group \(\Gamma \) . The classical Bochner–Schoenberg–EberleinBochner–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].