Injective envelopes for locally C*-algebras

TL;DR

This work extends Hamana's injective envelope theory to the realm of locally $C^{\ast}$-algebras by introducing admissible injective envelopes for unital Fréchet locally $C^{\ast}$-algebras and morphisms given by unital admissible local $\mathcal{CP}$-maps. The authors develop a framework of minimal admissible projections and seminorms, proving that every object possesses a unique admissible injective envelope up to a local isometric $\ast$-isomorphism, and that envelopes arise as the range of these minimal projections. A key tool is the Arens-Michael decomposition, which allows the envelope to be realized as the inverse limit of injective envelopes at each level $\mathcal{A}_{n}$; this yields concrete descriptions and stability results for the envelope. Moreover, for Fréchet locally $W^{\ast}$-algebras, injectivity is characterized by the componentwise injectivity of the $\mathcal{A}_{n}$, linking global injectivity to the inverse-system structure. The results provide a comprehensive extension of injective envelope theory to the locally convex setting and give structural insight via inverse-limit representations.

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 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*-algebras is injective if and only if the C*-algebras from its Arens-Michael decomposition are injective.

Theorems & Definitions (60)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Remark 2.6
• Definition 2.7
• Remark 2.8
• Lemma 2.9