[Paper Review] On Well-behaved Unbounded Representations of *-Algebras

Konrad Schmuedgen|ArXiv.org|Apr 26, 2000

Advanced Operator Algebra Research15 references21 citations

TL;DR

This paper introduces a general framework for constructing well-behaved unbounded *-representations of *-algebras using compatible pairs (X, B), where X is a *-algebra and B is a normed *-algebra with a compatible left action. For any continuous non-degenerate *-representation ρ of B, a closed *-representation ρ′ of X is canonically defined via ρ′(x)ρ(b) = ρ(x▹b), yielding a systematic method to generate physically and mathematically meaningful representations, particularly applicable to polynomial algebras, Lie algebras, pseudodifferential operators, and quantum groups.

ABSTRACT

A general approach to the well-behaved unbounded *-representations of a *-algebra X is proposed. Let B be a normed *-algebra equipped with a left action |> of X on B such that (x |> a)^+ b=a^+(x^+ |> b) for a,b\in B and x\in X. Then the pair (X,B) is called a compatible pair. For any continuous non-degenerate *-representation ρof B there exists a closed *-representation ρ' of X such that ρ'(x)ρ(b)=ρ(x |> b), where x\in X and b\in B. The *-representations ρ' are called the well-behaved *-representations associated with the compatible pair (X,B). A number of examples are developed in detail.

Motivation & Objective

• To address the fundamental problem that classifying all *-representations of general *-algebras is ill-posed due to excessive pathological examples.
• To identify and formalize a class of 'well-behaved' unbounded *-representations that exclude pathological behavior and are suitable for physical and mathematical applications.
• To provide a general, systematic method to construct such well-behaved representations using compatible pairs (X, B) of *-algebras with a compatible left action.
• To demonstrate the framework's applicability across diverse settings, including polynomial algebras, Lie group representations, pseudodifferential operators, and quantum groups.
• To clarify that well-behavedness is not absolute but depends on the algebraic context, and to unify various existing notions under a single formalism.

Proposed method

• Define a compatible pair (X, B) where X is a *-algebra, B is a normed *-algebra, and X acts on B via a left action ▹ satisfying the compatibility condition (x▹a)+b = a+(x+▹b) for all a,b ∈ B, x ∈ X.
• Use any continuous, non-degenerate *-representation ρ of B to induce a closed *-representation ρ′ of X via the formula ρ′(x)ρ(b) = ρ(x▹b).
• Prove that the induced map ρ′ is a well-defined closed *-representation of X, ensuring that the action is compatible with the *-algebra structure.
• Show that the construction is canonical and functorial in nature, preserving the algebraic and topological properties of the representations.
• Illustrate the framework through concrete examples: polynomial algebras, Lie algebra representations via C∞₀(G), Weyl calculus of pseudodifferential operators, and quantum groups like SUq(1,1).
• Demonstrate that in many cases, the action ▹ arises naturally as multiplication in a larger *-algebra A containing X and B as subalgebras, with ▹(x,b) = xb.

Experimental results

Research questions

• RQ1 How can one systematically define and construct 'well-behaved' unbounded *-representations of *-algebras, avoiding the ill-posedness of classifying all possible representations?
• RQ2 What algebraic structure ensures that a left action of a *-algebra X on a normed *-algebra B induces a closed *-representation of X from a representation of B?
• RQ3 In what ways do existing notions of well-behavedness—such as self-adjointness, essential self-adjointness, or the Weyl relation—fit into a unified framework?
• RQ4 How can the framework be applied to diverse algebras such as polynomial algebras, enveloping algebras of Lie algebras, and quantum algebras?
• RQ5 What is the role of the compatibility condition (x▹a)+b = a+(x+▹b) in ensuring the consistency and closure of the induced representation ρ′?

Key findings

• For any compatible pair (X, B) and continuous non-degenerate *-representation ρ of B, there exists a unique closed *-representation ρ′ of X such that ρ′(x)ρ(b) = ρ(x▹b) for all x ∈ X, b ∈ B.
• The construction generalizes and unifies various known classes of well-behaved representations, including those based on spectral projections, Weyl relations, and integrability conditions.
• In the case of the polynomial algebra C[x₁,…,xₙ], the framework yields well-behaved representations where the generators are essentially self-adjoint and their spectral projections commute.
• For the enveloping algebra E(g) of a Lie algebra g, the framework recovers G-integrable representations via the action of X on C∞₀(G) with convolution multiplication.
• For the quantum plane O(R²q), the Weyl calculus of pseudodifferential operators provides a natural compatible pair, yielding well-behaved representations of the coordinate *-algebra.
• The framework explains why many physical and mathematical papers implicitly assume well-behavedness: such assumptions are necessary to avoid pathological representations and ensure physical consistency.