Injective envelopes for locally C*-algebras

Abstract

We introduce the notion of admissible injective envelope for a locally C*-algebra and show that each object in the category whose objects are unital Fr\'{e}chet locally C*-algebras and whose morphisms are unital admissible local completely positive maps has a unique admissible injective envelope. The concept of admissible injectivity is stronger than that of injectivity. As a consequence, we show that a unital Fr\'{e}chet locally W*-algebras is injective if and only if the C*-algebras from its Arens-Michael decomposition are injective.