Banach Function Algebras, Arens Regularity, and BSE Norms

In this first chapter, we shall recall some notation and preliminary results within the functional analysis and related subjects that we shall use.

In this chapter, we shall introduce Banach algebras and recall their basic properties. General Banach algebras are considered in \(\S 2.1\), and then we shall turn to a special case, that of \(C^*\)-algebras and von Neumann algebras, in \(\S 2.2\). Our major theme, to be commenced in \(\S 2.3\), will be consideration of the bidual space \(A''\) of a Banach algebra A and of the two Arens products, \(\Box \) and \(\Diamond \), that are defined on the Banach space \(A''\), each making \(A''\) into a Banach algebra that contains A as a closed subalgebra. The Banach algebra A is ‘Arens regular’ if these two products coincide on \(A''\). In \(\S 2.3\), we shall give various examples of Arens regular Banach algebras and of Banach algebras that are not Arens regular; for example, every \(C^*\)-algebra A is Arens regular, and \((A'', \,\Box \,)\) is itself a von Neumann algebra, called the ‘enveloping von Neumann algebra’. However, the group algebra \((\ell ^{\,1}(G), \,\star \,)\) of a group G is not Arens regular whenever G is infinite. These ideas will be substantially developed in Chapter 6. We shall also consider when a Banach algebra is an ideal in its bidual.

In this chapter, we shall consider the classical Banach algebras of harmonic analysis that are associated with a locally compact group G. The class of these algebras includes the group algebra \(L^1(G)\), the measure algebra M(G), the related algebras \(L^p(G)\) (which are Banach algebras when G is compact and \(1\le p\le \infty \)), and Beurling algebras on semigroups and locally compact groups. The product in each of these algebras is given by convolution, denoted by \(\,\star \,\). For example, \((M(G),\,\star \,)\) is an isometric dual Banach algebra, with Banach-algebra predual \(C_{\,0}(G)\), for each locally compact group G. The definitions and some basic properties of these algebras will be recalled in \(\S 4.1\).

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].