Gelfand [1]1 has shown that a real Banach algebra in which for every elementwe have ||x2|| = ||x||2, is isomorphic and isometric to the ring continuousfunctions on some compact Hausdorff space. Since he was concerned with an abstractBanach algebra, his representation for this space is necessarily quitecomplicated; indeed, it is in terms of a space of maximal ideals of the Banachalgebra. One would expect, then, that for a particular Banach algebra a simplercharacterization of this space would be obtained. It is the purpose of this paperto find such a simpler representation for the collection of Baire functions of class a, for each a ≥, over a topological space S. These collections satisfy the conditions of Gelfand's theorem. Our representation, which is done in termsof lattice, instead of ring, operations, will give the space as a Boolean space associated with a Boolean algebra of subsets of the original space S.
The paper is divided into two parts. In part I, we define the Baire functionsof class a and obtain some results connecting them and the Boolean algebra. Part II is concerned with the representation theorem, some of its consequences, and examples to show that the theory is non-vacuous.
1.References to the literature are indicated by numbers in square brackets.
