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

💡 Research Summary

The paper extends Hamana’s theory of injective envelopes from the category of unital C*‑algebras with unital completely positive (UCP) maps to the broader setting of locally C*‑algebras, focusing on the subcategory whose objects are unital Fréchet locally C*‑algebras and whose morphisms are unital admissible local completely positive (ALCP) maps.

A locally C*‑algebra A is a complete Hausdorff ‑algebra whose topology is generated by an upward‑filtered family of C‑seminorms {pλ}λ∈Λ. For each λ the quotient Aλ = A / ker pλ is a genuine C*‑algebra, and A can be identified with the inverse limit lim←Aλ; this is the Arens‑Michael decomposition. The paper first recalls the basic notions of positivity, local positivity, and local complete positivity in this context, emphasizing that a linear map φ: A→B is locally completely positive (LCP) if for each seminorm on B there exists a seminorm on A making φ positive at the corresponding level.

The authors then introduce a stricter class of maps: admissible local completely positive (ALCP) maps. An ALCP map requires that the index sets of source and target seminorms coincide (Δ = Λ) and that positivity be preserved at each identical level λ. This condition is stronger than ordinary LCP but weaker than the R‑injectivity condition previously studied by Dosiev.

With ALCP maps as morphisms, the paper defines an object I to be admissibly injective if every unital ALCP map defined on a unital self‑adjoint subspace S⊂B (containing the unit) extends to a unital ALCP map on the whole algebra B. This mirrors the classical definition of injectivity but now respects the finer local structure.

The central results are twofold. First, existence: for any unital Fréchet locally C*‑algebra A, the authors construct a minimal family of B‑seminorms and a corresponding admissible projection eA: A→A (an idempotent ALCP map). The range C*(eA) equipped with the induced multiplication and involution becomes a unital locally C*‑algebra, denoted IA, which serves as the admissible injective envelope of A. The construction relies on a careful analysis of B‑seminorms, admissible B‑projections, and the Schwarz inequality for ALCP maps.

Second, uniqueness: if (B1,Φ1) and (B2,Φ2) are two admissible injective envelopes of A, there exists a unique local isometric *‑isomorphism Ψ: B1→B2 satisfying Ψ∘Φ1 = Φ2. The proof adapts Hamana’s argument, using the fact that the envelope is the range of an admissible projection and that ALCP maps are automatically completely positive and continuous on each level.

A further major contribution is the description of IA as an inverse limit of the classical injective envelopes of the C*‑components Aλ. Since each Aλ is a C*‑algebra, Hamana’s theory provides its injective envelope I(Aλ). The paper shows that IA ≅ lim← I(Aλ) as locally C*‑algebras, and consequently A is admissibly injective if and only if every Aλ is injective. This bridges the local theory with the well‑understood global C*‑theory.

The authors apply this framework to Fréchet locally W*‑algebras (inverse limits of W*‑algebras). In this setting, admissible injectivity, R‑injectivity, and ordinary injectivity coincide. Hence a unital Fréchet locally W*‑algebra is injective precisely when each W*‑component in its Arens‑Michael decomposition is injective. This recovers and extends results of Dosiev concerning the bounded part b(A) and its injectivity.

Technical highlights include:

• A local version of the Schwarz inequality for ALCP maps, establishing that φ(a)φ(a) ≤ φ(a a) holds at each seminorm level.
• The proof that the image of an admissible projection inherits a family of C*‑seminorms, making it a genuine locally C*‑algebra.
• Lemma 2.4, which shows that a bijective unital map whose forward and inverse are ALCP is automatically a local isometric *‑isomorphism.
• Detailed handling of continuity issues: while positivity in C*‑algebras forces continuity, this is not automatic in the locally C* setting; the paper shows that LCP maps are continuous, and ALCP maps are even stronger.

In summary, the paper establishes a robust theory of admissible injective envelopes for unital Fréchet locally C*‑algebras, proving existence, uniqueness, and a concrete description via inverse limits of classical injective envelopes. The results deepen the understanding of injectivity in the non‑normed operator algebra world and provide tools for further investigations of non‑commutative topological algebras, quantum functional analysis, and the structure of infinite‑dimensional operator systems.