Lie Algebras of Derivations and Resolvent Algebras

TL;DR

This paper explores how Lie algebra actions by derivations on C*-algebras relate to resolvent algebras, revealing connections to symplectic structures and the algebraic framework of quantum mechanics.

Contribution

It demonstrates that under certain conditions, the algebra generated by pseudo-resolvents aligns with the resolvent algebra of a symplectic space, linking derivation actions to quantum algebra structures.

Findings

Pseudo-resolvents affiliated with derivations satisfy an 'almost inner' property.

In the abelian case, a central extension determines pseudo-resolvent relations.

Ergodic actions induce a symplectic form that defines the resolvent algebra.

Abstract

This paper analyzes the action {\delta} of a Lie algebra X by derivations on a C*-algebra A. This action satisfies an "almost inner" property which ensures affiliation of the generators of the derivations {\delta} with A, and is expressed in terms of corresponding pseudo-resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*-algebra A, it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo- resolvents. If the Lie action {\delta} is ergodic, i.e. the only elements of A on which all the derivations in {\delta}_x vanish are multiples of the identity, then this extension is given by a (non-degenerate) symplectic form {\sigma} on X. Moreover, the algebra generated by the pseudo-resolvents coincides with the resolvent algebra based on the symplectic space (X, {\sigma}). Thus the resolvent algebra of the canonical commutation relations, which was recently introduced in physically motivated analyses of quantum systems, appears also naturally in the representation theory of Lie algebras of derivations acting on C*-algebras.