BISHOP AND SILOV DECOMPOSITIONS: A STUDY ON VECTOR FUNCTION SPACES

Abstract

This paper has been devoted to the study of vector function spaces on X. We define the Bishop and Silov decompositions in several ways for vector function spaces. If A denotes a complex function space on X then the tensor product A $ B of A and Banach algebra B can be regarded as a vector function space on X. The concept of the slice product A B of a function algebra A with a Banach algebra B has been defined earlier [1]. We extend the idea of A # B for a complex function space A on X. Then A # B also can be considered as a vector function space on X and A B  A # B, Mainly, we concentrate our study to the decompositions and their properties for vector function spaces of the types A B and A # B, where A denotes a complex function space on X. In particular, we show that the Bishop (Silov) decompositions for a complex function space A and the vector function space A B are the same.