COMMUTATIVE GELFAND THEORY FOR REAL BANACH-ALGEBRAS - REPRESENTATIONS AS SECTIONS OF BUNDLES

We are concerned here with the development of a more general real case of the classical theorem of Gelfand ([5], 3.1.20), which represents a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. In sectional sign 1 we point out that when looking at real algebras there is not always a one-to-one correspondence between the maximal ideals of the algebra B, denoted M(B), and the set of unital (real) algebra homomorphisms from B into C, denoted by phi(B). This simple point and subsequent observations lead to a theory of representations of real commutative unital Banach algebras where elements are represented as sections of a bundle of real fields associated with the algebra (Theorem 3.5). After establishing this representation theorem, we look into the question of when a real commutative Banach algebra is already complex. There is a natural topological obstruction which we delineate. Theorem 4.8 gives equivalent conditions which determine whether such an algebra is already complex. Finally, in sectional sign 5 we abstractly characterize those section algebras which appear as the target algebras for our Gelfand transform. We dub these algebras "almost complex C*-algebras" and provide a natural classification scheme.