Banach algebras

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.