Injective envelopes for locally C *-algebras*

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 Fréchet 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 Fréchet locally W∗-algebra is injective if and only if the W∗-algebras from its Arens-Michael decomposition are injective.