On the boundary of some function algebras

The aim of this thesis is to prove the existence of the Shilov boundary and the minimal boundary with respect to some function algebras and investigate their topological structures.Science, Faculty ofMathematics, Department ofGraduat

On certain rings of e-valued continuous functions

Let C(X,E) denote the set of all continuous functions from a topological space X into a topological space E. R. Engelking and S. Mrowka [2] proved that for any E-completely regular space X [Definition 1.1], there exists a unique E-compactification [formula omitted] [Definitions 2.1 and 3.1] with the property that every function f in C(X,E) has an extension f in [formula omitted]. It is proved that if E is a (*)-topological division ring [Definition 5-5] and X is an E-completely regular space, then [formula omitted] is the same as the space of all E-homomorphisms [Definition 5.3] from C(X,E) into E. Also, we establish that if E is an H-topological ring [Definition 6.1] and X, Y are E-compact spaces [Definition 2.1], then X and Y are homeomorphic if, and only if, the rings C(X,E) and C(Y,E) are E-isomorphic [Definition 5.3]. Moreover, if t is an E-isomorphism from C(X,E) onto C(Y,E) then [formula omitted] is the unique homeomorphisms from Y onto X with the property that [formula omitted] for all f in C(X,E), where π is the identity mapping on X and t is a certain mapping induced by t. In particular, the development of the theory of C(X,E) gives a unified treatment for the cases when E is the space of all real numbers or the space of all integers. Finally, for a topological ring E, the bounded subring C*(X,E) of C(X,E) is studied. A function f in C(X,E) belongs to C*(X,E) if for any O-neighborhood U in E, there exists a 0-neighborhood V in E such that f[X]•V c U and V•f[X] c U. The analogous results for C*(X,E) follow closely the theory of C(X,E); namely, for any E*-completely regular space X [Definition 9.5], there exists an E*-compactification [formula omitted] of X such that every function f in C (X,E) has an extension f in [formula omitted] when E is the space of all nationals, real numbers, complex numbers, or the real quaternions, [formula omitted] is just the space of all E-homomorphisms from C*(X,E) into E. This is also valid for a topological ring E which satisfies certain conditions. Also, two E*-compact spaces [Definition 10.1] X and Y are homeomorphic if, and only if, the rings C*(X,E) and C*(Y,E) are E-isomorphic, where E is any H*-topological ring [Definition 12.8].Science, Faculty ofMathematics, Department ofGraduat