Maximal ideal space of function algebras

We present a representation theory for the maximal ideal space of a real function algebra, endowed with the Gelfand topology, using the theory of uniform spaces. Application are given to algebras of differentiable functions in a normęd space, improving and generalizing some known results.

SEPARATION PROPERTIES AND OPERATING FUNCTIONS ON A SPACE OF CONTINUOUS FUNCTIONS

Characterizations of the space CR (X) of all real-valued continuous functions on a compact Hausdorff space X among its subspaces are investigated under the circumstances of operating functions. One of the main purpose in this paper is to disprove the following conjecture: if a non-affine function operates on an ultraseparating real Banach function space E on X, then E = CR (X). A positive answer is given in the case that E satisfies a stronger separation axiom than ultraseparation one, which the real part of an ultraseparating Banach function algebra satisfies. For the original conjecture a counterexample is given; there is an ultraseparating real Banach function space on a compact metric space Y on which the function |·| operates, but it does not coincide with CR (Y). A characterization is given for non-affine functions which operate only on CR (X) among ultraseparating real Banach function spaces on X. By using these results the symbolic calculus on real Banach function spaces is investigated without extra hypothesis of ultraseparation. Several non-local-Lipschitz functions are shown not to operate on a real Banach function space E on X unless E = CR (X). In particular, the function tp defined on [0,1) for a p with 0 < p < 1 or the standard Cantor function on [0, 1] never operates on a real Banach function space E on X unless E = CR (X). Functions which operate on the real part of a real function algebra are also investigated. A positive answer is given for the conjecture that only affine functions operate on the real part of a non-trivial real function algebra.

Choquet Boundary for Real Function Algebras

The concepts of Choquet boundary and Shilov boundary are well-established in the context of a complex function algebra (see [2] for example). There have been a few attempts to develop the concept of a Shilov boundary for real algebras, [4], [6] and [7]. But there seems to be none to develop the concept of Choquet boundary for real algebras.The aim of this paper is to develop the theory of Choquet boundary of a real function algebra (see Definition (1.8)) along the lines of the corresponding theory for a complex function algebra.In the first section we define a real-part representing measure for a continuous linear functional ϕ on a real function algebra A with the property ║ϕ║ = 1 = ϕ(1). The elements of A are functions on a compact, Hausdorff space X. The Choquet boundary is then defined as the set of those points x ∊ X such that the real part of the evaluation functional, Re(ex), has a unique real part representing measure.

A Topological Characterization of Gleason Parts of Real Function Algebras

It is well-known that a topological space is a Gleason part of some complex function algebra if and only if it is completely regular and σ-compact. In the present paper, a Gleason part of a real function algebra is characterized as a completely regular σ-compact topological space which admits an involutoric homeomorphism.