[Paper Review] Morphisms of certain Banach C*-modules

Фабио Багарелло, Camillo Trapanı|ArXiv.org|Apr 6, 2009

Advanced Banach Space Theory4 citations

TL;DR

This paper introduces two distinct generalizations of the Gelfand-Naimark-Segal (GNS) construction for CQ*-algebras—Banach C*-modules that bridge C*-algebras and quasi*-algebras—by using positive linear functionals with admissibility conditions and positive sesquilinear forms with invariance properties. The key contribution is a representation theory for CQ*-algebras via bimorphisms, yielding concrete operator realizations in Hilbert space triples.

ABSTRACT

Morphisms and representations of a class of Banach C*-modules, called CQ*algebras, are considered. Together with a general method for constructing CQ*-algebras, two different ways of extending the GNS-representation are presented.

Motivation & Objective

• To develop a representation theory for CQ*-algebras, a class of Banach C*-modules generalizing C*-algebras.
• To address the lack of concrete operator realizations for abstract CQ*-algebras, especially in the context of partial *-algebras.
• To extend the GNS construction to CQ*-algebras through two distinct, non-equivalent approaches.
• To establish conditions under which morphisms and representations of CQ*-algebras yield bounded operators on Hilbert spaces.
• To clarify the role of *-semisimple CQ*-algebras and their structure via functional calculus and invariance properties.

Proposed method

• Constructs CQ*-algebras via a norm-completion method starting from a C*-algebra with a weaker, isometric involution-compatible norm.
• Defines morphisms as bimorphisms to ensure compatibility with the algebraic and topological structure of CQ*-algebras.
• Introduces the first GNS-type construction using a positive linear functional satisfying admissibility conditions (s1)–(s4).
• Proposes a second GNS-type construction based on a positive sesquilinear form Ω satisfying (s1)–(s5), including invariance and boundedness conditions.
• Establishes a triplet of Hilbert spaces H_♭ ⊂ H₀ ⊂ H_♯, where the representation π maps A ∈ A into bounded operators from H_♭ to H_♯.
• Uses the Cauchy-Schwarz inequality and the invariance condition (s5) to prove that π(A*) = π(A)*, ensuring *-representation structure.

Experimental results

Research questions

• RQ1 Can the GNS construction be generalized to CQ*-algebras, which are non-unital and partial *-algebras?
• RQ2 What conditions on a positive linear functional or sesquilinear form ensure the existence of a bounded representation on a Hilbert space?
• RQ3 How do the two proposed GNS-type constructions differ in their foundational assumptions and resulting representations?
• RQ4 What role does the bimorphism concept play in defining morphisms for CQ*-algebras?
• RQ5 Under what conditions does a sesquilinear form on a CQ*-algebra yield a *-representation via operator realization?

Key findings

• A new class of CQ*-algebras is constructed via completion of a C*-algebra under a weaker, isometrically involution-compatible norm.
• The notion of bimorphism is identified as the natural morphism for CQ*-algebras, ensuring compatibility with the module and involution structure.
• The first GNS-type construction is based on a positive linear functional satisfying admissibility and boundedness conditions, yielding a representation into B(H_♭, H_♯).
• The second GNS-type construction uses a positive sesquilinear form Ω satisfying (s1)–(s5), including invariance under involution and a growth control condition (s4), to define a *-representation.
• The representation π satisfies π(AX) = π(A)π_♭(X) and π(A*) = π(A)*, confirming that π is a *-representation when (s5) holds.
• A sesquilinear form satisfying (s1)–(s5) and Ω(𝕀,𝕀) = 1 induces a state ω(A) = Ω(A,𝕀) that satisfies the standard GNS conditions, linking the two constructions.