Injective hulls of 𝐶* algebras

Abstract

Introduction. Our aim is to apply category concepts to the study of commutative C* algebras. In particular we shall study injective hulls. This appears to have been overlooked in the literature, although injective Banach spaces have been studied extensively. However, in [2] the dual category of compact Hausdorff spaces is studied. It is shown that the projectives are precisely the extremely disconnected spaces and that projective covers always exist. Hence, injective hulls always exist in the original category. Injective hulls are very important in ring theory, where they are used to obtain generalized quotient rings. They play a critical role in the proof of the Mitchell embedding theorem for abelian categories. They lead also to a novel way of obtaining the decomposition theorem for commutative noetherian rings. Recently Isbell constructed injective hulls in the category of metric spaces [5]. Injective hulls in Banach spaces are discussed in [1] and [4].