Countably-generated intermediate algebras between C^*(X) and C(X)
by
Javier Gómez Pérez
Departamento de Matemáticas. Universidad de León. León. Spain
Coauthors: Jesús M. Domínguez (Universidad de Valladolid, Spain)

Let X be a completely regular space, and A an intermediate algebra between C^*(X) and C(X). We shall briefly say that A is an intermediate algebra on X. Any one of these algebras can be obtained by adjoining to C^*(X) a suitable family of functions in C(X). We study those intermediate algebras on X that are obtained by adjoining a countable family of functions in C(X), which we shall call countably-generated intermediate algebras.

Taking into account the order properties of the intermediate algebras, it is shown that the countably-generated intermediate algebras are precisely those intermediate algebras on X that contain a countable cofinal subset.

Algebraic properties as that a countably-generated intermediate algebras A is not C-type nor closed under countable composition follows from some results of Gillman-Jerison and Henriksen-Isbell-Johnson respectively, and both are consequences of the existence of a countable cofinal subset in the quotient ring of A by an hyper-real maximal ideal. But the above argument is not sufficient to assure that A is not closed under composition. In spite of that, we show that the countably-generated intermediate algebras are not closed under composition.

An example of an intermediate algebra on N previously studied by R.M. Brooks and D. Plank, that is not closed under composition, is examined in order to show that it is not a countably-generated intermediate algebra.

Date received: June 2, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-36.