Abstract
This paper analyzes the action δ of a Lie algebra X by derivations on a C*–algebra \({\mathcal{A}}\) . This action satisfies an “almost inner” property which ensures affiliation of the generators of the derivations δ with \({\mathcal{A}}\) , and is expressed in terms of corresponding pseudo–resolvents. In particular, for an abelian Lie algebra X acting on a primitive C*–algebra \({\mathcal{A}}\) , it is shown that there is a central extension of X which determines algebraic relations of the underlying pseudo–resolvents. If the Lie action δ is ergodic, i.e. the only elements of \({\mathcal{A}}\) on which all the derivations in δ X vanish are multiples of the identity, then this extension is given by a (non–degenerate) symplectic form σ on X. Moreover, the algebra generated by the pseudo–resolvents coincides with the resolvent algebra based on the symplectic space (X, σ). 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aczél J.: The general solution of two functional equations by reduction to functions additive in two variables and with the aid of a Hamel basis. Glasnik Mat.–Fiz. Astronom. 20, 65–73 (1965)
Buchholz D., Grundling H.: The resolvent algebra: A new approach to canonical quantum systems. J. Funct. Anal. 254, 2725–2779 (2008)
Buchholz D., Grundling H.: Algebraic supersymmetry: A case study. Commun. Math. Phys. 272, 699–750 (2007)
Dixmier, J.: C*-algebras, Amsterdam: North Holland, 1977
Kato T.: Remarks on pseudo-resolvents and infinitesimal generators of semi-groups. Proc. Japan Acad. 35, 467–468 (1959)
Sakai, S.: C*–algebras and W*–algebras. Berlin-Heidelberg-New York: Springer, 1971
Yosida, K.: Functional Analysis, Berlin-Heidelberg-New York: Springer, 1980
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Y. Kawahigashi
Dedicated to the memory of Hans-Juergen Borchers
Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
Rights and permissions
About this article
Cite this article
Buchholz, D., Grundling, H. Lie Algebras of Derivations and Resolvent Algebras. Commun. Math. Phys. 320, 455–467 (2013). https://doi.org/10.1007/s00220-012-1567-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1567-z