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.