## Lie Algebras of Derivations and Resolvent Algebras    [PDF]

Detlev Buchholz, Hendrik Grundling
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.
View original: http://arxiv.org/abs/1202.2780