The Representation of Baire Functions
<p>Gelfand [1]<sup>1</sup> has shown that a real Banach algebra in which for every element we have ||x<sup>2</sup>|| = ||x||<sup>2</sup>, is isomorphic and isometric to the ring continuous functions on some compact Hausdorff space. Since he was concerned w...
| Summary: | <p>Gelfand [1]<sup>1</sup> has shown that a real Banach algebra in which for every element
we have ||x<sup>2</sup>|| = ||x||<sup>2</sup>, is isomorphic and isometric to the ring continuous
functions on some compact Hausdorff space. Since he was concerned with an abstract
Banach algebra, his representation for this space is necessarily quite
complicated; indeed, it is in terms of a space of maximal ideals of the Banach
algebra. One would expect, then, that for a particular Banach algebra a simpler
characterization of this space would be obtained. It is the purpose of this paper
to 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 terms
of 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.</p>
<p>The paper is divided into two parts. In part I, we define the Baire functions
of 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.</p>
<p>1. References to the literature are indicated by numbers in square brackets.</p>
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