[Paper Review] Extension of representations in quasi *-algebras

Jean-Pierre Antoine, Фабио Багарелло+1 more|ArXiv.org|Apr 6, 2009

Advanced Algebra and Logic18 references3 citations

TL;DR

This paper develops a closure-based extension method for *-representations of the dense *-subalgebra 𝔄₀ in a topological quasi *-algebra 𝔄, enabling the extension of unbounded operators beyond the original domain. The key contribution is showing that representations can be extended via limits in the s*-topology, with explicit realization in the CCR algebra and L^p-L^s duality, yielding a larger quasi *-algebra of operators in the dual space.

ABSTRACT

Let $(A, A_o)$ be a topological quasi *-algebra, which means in particular that $A_o$ is a topological *-algebra, dense in $A$. Let $π^o$ be a *-representation of $A_o$ in some pre-Hilbert space ${\cal D} \subset {\cal H}$. Then we present several ways of extending $π^o$, by closure, to some larger quasi *-algebra contained in $A$, either by Hilbert space operators, or by sesquilinear forms on ${\cal D}$. Explicit examples are discussed, both abelian and nonabelian, including the CCR algebra.

Motivation & Objective

• To address the challenge of extending *-representations of a dense *-subalgebra 𝔄₀ in a topological quasi *-algebra 𝔄 beyond the original domain.
• To develop a closure-based extension procedure that generates new operators not in the original representation, differing from standard closure of unbounded operators.
• To establish conditions under which a representation on 𝔄₀ can be extended to a larger quasi *-algebra within 𝔄 via convergence in the s*-topology.
• To provide explicit realizations of the extension process in physically relevant examples, including the CCR algebra and L^p-L^s duality.
• To clarify the role of topologies (especially s*-topology) in enabling the extension of representations defined on unbounded domains.

Proposed method

• Define a representation π⁰ of 𝔄₀ on a dense subspace 𝒟 ⊂ 𝒫, with values in the partial O*-algebra 𝒫(𝒟,𝒫) of operators mapping 𝒟 into 𝒫 and their adjoints into 𝒟.
• Introduce the s*-topology on 𝒫(𝒟,𝒫) using seminorms ‖Aφ‖ and ‖A†φ‖ for φ ∈ 𝒟, which is weaker than the uniform topology.
• Define the set 𝔸̃(π, τ_op) as the set of elements X ∈ 𝔄 for which there exists a net {Xα} ⊂ 𝔄₀ such that Xα → X in τ and π⁰(Xα) is τ_op-Cauchy.
• Extend π⁰ to X ∈ 𝔸̃(π, τ_op) by setting π(X) = τ_op-lim π⁰(Xα), with values in the τ_op-completion of 𝒫(𝒟,𝒫).
• Use duality between L^p(X) and L^s(X) with s = 2p/(p−2) to show that 𝔸̃(π, τ_s*) = L^s(X) when π⁰ is the GNS representation on L^p(X).
• Demonstrate that π⁰ is τ_s*-extendible by showing that if fₙ → 0 in L¹ and π⁰(fₙ) → Y in s*-topology, then Y = 0, ensuring uniqueness of the limit.

Experimental results

Research questions

• RQ1 Can a *-representation π⁰ of a dense *-subalgebra 𝔄₀ in a topological quasi *-algebra 𝔄 be extended to a larger algebra of unbounded operators?
• RQ2 What topological structure on the representation space enables such an extension, and how does it differ from standard closure of unbounded operators?
• RQ3 In the case of the CCR algebra or L^p-L^s duality, what is the precise domain of the extended representation?
• RQ4 How does the s*-topology facilitate the extension of representations that are not uniformly bounded?
• RQ5 Under what conditions is the extended representation unique and well-defined in the completion of the representation space?

Key findings

• The representation π⁰ on C(X) ⊂ L¹(X) is not τ_op-extendible to L^s(X) under the uniform topology, as it is already closed in the C*-norm.
• However, under the s*-topology, π⁰ is τ_s*-extendible, and the extended domain is precisely L^s(X), where s = 2p/(p−2).
• The s*-topology completion of 𝒫(𝒟,𝒫) is the full partial O*-algebra 𝒫†(𝒟,𝒫), showing that s*-closure captures all possible extensions.
• For the GNS representation on L^p(X), the set of τ_s*-extendible elements is exactly L^s(X), with s = 2p/(p−2), via duality with L^p(X).
• The extension process yields a larger quasi *-algebra 𝔸̃(π, τ_s*) ⊂ 𝔄, containing 𝔄₀ and closed under multiplication and involution.
• The construction provides a concrete realization of the extension of the CCR algebra via L^s(X) as the extended observable algebra.