The resolvent algebra: Ideals and dimension

TL;DR

This paper investigates the mathematical structure of the resolvent algebra associated with symplectic spaces, revealing its nuclearity, type I classification in finite dimensions, and detailed ideal structure.

Contribution

It provides a comprehensive analysis of the ideal structure of the resolvent algebra, including classifications of maximal, minimal, and principal ideals, and establishes its nuclearity and type I properties.

Findings

R(X,σ) is always nuclear.

In finite dimensions, R(X,σ) is of type I.

Principal ideals form the building blocks of all ideals.

Abstract

Let (X,\sigma) be a symplectic space admitting a complex structure and let R(X,\sigma) be the corresponding resolvent algebra, i.e. the C*-algebra generated by the resolvents of selfadjoint operators satisfying canonical commutation relations associated with (X,\sigma). In previous work this algebra was shown to provide a convenient framework for the analysis of quantum systems. In the present article its mathematical properties are elaborated with emphasis on its ideal structure. It is shown that R(X,\sigma) is always nuclear and, if X is finite dimensional, also of type I (postliminal). In the latter case dim(X) labels the isomorphism classes of the corresponding resolvent algebras. For X of arbitrary dimension, principal ideals are identified which are the building blocks for all other ideals. The maximal and minimal ideals of the resolvent algebra are also determined.