[Paper Review] Abelian topological groups with host algebras

Hendrik Grundling, Karl‐Hermann Neeb|arXiv (Cornell University)|May 16, 2006

Advanced Operator Algebra Research6 citations

TL;DR

This paper constructs a C*-algebra using Blackadar’s infinite tensor product of nonunital C*-algebras to model the canonical commutation relations (CCRs) on a countably infinite-dimensional symplectic space (S,B). The resulting algebra serves as a host algebra for the σ-representation theory of the abelian group S, ensuring its representation space consists precisely of regular representations, thereby enabling direct integral decompositions into irreducible regular representations.

ABSTRACT

The Weyl algebra,- the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S,B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a host (i.e. a generalised group algebra, explained below) for the \sigma-representation theory of the abelian group S where \sigma(.,.):=e^{iB(.,.)/2}. As an easy application, it then follows that for every regular representation of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result). An Erratum for this paper is added at the end.

Motivation & Objective

• To resolve the defect in the standard Weyl algebra, which includes non-regular representations not suitable for physical field models.
• To construct a C*-algebra whose representation space fully captures only the regular representations of the CCRs on a countably infinite-dimensional symplectic space.
• To establish a host algebra for the σ-representation theory of the abelian group S, where σ(x,y) = exp(iB(x,y)/2).
• To provide a rigorous algebraic framework that ensures every regular representation on a separable Hilbert space admits a direct integral decomposition into irreducible components.

Proposed method

• Utilizes Blackadar’s version of infinite tensor products of nonunital C*-algebras to construct the target C*-algebra.
• Defines the C*-algebra as a host algebra for the σ-representation theory of the abelian group S, with σ(x,y) = exp(iB(x,y)/2).
• Constructs the algebra in such a way that its representations correspond exactly to the regular representations of the CCRs on (S,B).
• Employs the structure of the symplectic space (S,B) to parameterize the commutation relations via the exponential factor exp(iB(x,y)/2).
• Applies the theory of host algebras to ensure the representation space is precisely the set of regular representations.
• Demonstrates that every regular representation on a separable Hilbert space decomposes as a direct integral of irreducible regular representations.

Experimental results

Research questions

• RQ1 Can a C*-algebra be constructed such that its representations are exactly the regular representations of the CCRs on a countably infinite-dimensional symplectic space?
• RQ2 How can infinite tensor products of nonunital C*-algebras be used to realize a host algebra for σ-representations of an abelian group?
• RQ3 Does the constructed C*-algebra allow for a direct integral decomposition of every regular representation into irreducible components?
• RQ4 What structural properties must a C*-algebra possess to serve as a host algebra for the σ-representation theory of an abelian group?
• RQ5 Can the standard Weyl algebra’s defect—its inclusion of non-regular representations—be systematically eliminated via a new algebraic construction?

Key findings

• The constructed C*-algebra realizes a host algebra for the σ-representation theory of the abelian group S, with σ(x,y) = exp(iB(x,y)/2).
• The representation space of the C*-algebra consists precisely of the regular representations of the CCRs on the symplectic space (S,B).
• Every regular representation of the Weyl algebra on a separable Hilbert space admits a direct integral decomposition into irreducible regular representations.
• The construction relies on Blackadar’s infinite tensor product of nonunital C*-algebras to ensure the correct representation structure.
• The resulting algebra eliminates non-regular representations, resolving a known defect of the standard Weyl algebra.
• The method provides a canonical framework for modeling physical fields via only physically admissible representations.