[Paper Review] Derivations of quasi *-algebras

Фабио Багарелло, A. Inoue+1 more|ArXiv.org|Mar 31, 2009

Advanced Topics in Algebra7 references4 citations

TL;DR

This paper investigates the spatiality of derivations on quasi *-algebras, particularly focusing on whether a derivation defined on a dense *-subalgebra can be implemented by a linear operator. Using representation theory and topological closure techniques, the authors establish conditions under which a derivation arising as a limit of spatial derivations remains spatial, with the implementing operator emerging as a limit of the original implementing operators. The key contribution is a sufficient condition for the spatiality of such limit derivations in the context of physical systems with thermodynamic limits.

ABSTRACT

The spatiality of derivations of quasi *-algebras is investigated by means of representation theory. Moreover, in view of physical applications, the spatiality of the limit of a family of spatial derivations is considered.

Motivation & Objective

• To investigate the spatiality of derivations on quasi *-algebras, particularly in the context of physical systems with thermodynamic limits.
• To determine under what conditions a derivation defined on a dense *-subalgebra of a quasi *-algebra can be implemented by a linear operator.
• To analyze the limit of a net of spatial derivations and establish conditions under which the limit derivation remains spatial.
• To extend results to qu*-representations in rigged Hilbert spaces, relevant for quantum field theory and singular field operators.
• To provide a rigorous framework for defining algebraic dynamics in quantum systems via limit processes of regularized Hamiltonians.

Proposed method

• The paper uses representation theory to analyze derivations on quasi *-algebras, particularly focusing on *-representations and their extensions to larger domains.
• It introduces the notion of τ-closability to extend derivations beyond the dense *-subalgebra using topological completion under a locally convex topology τ.
• The authors define a qu*-representation of a *-algebra on a rigged Hilbert space, allowing for the treatment of singular operators such as point-like quantum fields.
• They establish joint continuity of sesquilinear forms associated with representations to ensure the well-definedness of extended operators and derivations.
• A key technique involves estimating norms via operators in the dual space of the dense domain, using a topology t† on the domain of the representation.
• The spatiality of the derived map is proven by showing that the extended derivation satisfies the commutator form δ(A) = [H, A] for some operator H.

Experimental results

Research questions

• RQ1 Under what conditions is a derivation on a quasi *-algebra spatial, i.e., implementable by a linear operator?
• RQ2 Can a derivation that arises as a limit of spatial derivations be spatial, and is the implementing operator the limit of the original implementing operators?
• RQ3 How can the spatiality of derivations be extended from a *-algebra to its completion under a locally convex topology τ?
• RQ4 What conditions ensure that a *-derivation induced by a qu*-representation in a rigged Hilbert space is spatial?
• RQ5 In the context of quantum field theory, when can a field's infinitesimal time translation be represented as a spatial derivation on a quasi *-algebra of observables?

Key findings

• A derivation δ on a quasi *-algebra (A, A₀) is spatial if it is τ-closable and the associated sesquilinear form is jointly continuous on the dense domain of a *-representation.
• The limit of a net of spatial derivations {δL} is spatial if the corresponding implementing operators {HL} converge in a suitable topology, ensuring the limit derivation is implemented by the limit operator H.
• The extension of a *-representation π₀ to the completion A is well-defined and (τ−τqs)-continuous, allowing the derivation δπ to be properly defined on A.
• The spatiality of the derived map δπ is established by showing that δπ(A) = [H, A] holds for all A in A, with H constructed from the limit of the implementing operators.
• The results are extended to qu*-representations in rigged Hilbert spaces, providing a framework for handling singular quantum fields as elements of L(D, D′).
• The paper provides a rigorous foundation for defining algebraic dynamics in quantum systems via thermodynamic limits, particularly when the Hamiltonian is regularized and then taken to the limit.