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On Lie algebras of operators
Journal article   Open access   Peer reviewed

On Lie algebras of operators

Ola Bratteli, George A Elliott, F. M Goodman and Palle E.T Jorgensen
Journal of functional analysis, Vol.86(2), pp.341-359
1989
DOI: 10.1016/0022-1236(89)90056-6
url
https://doi.org/10.1016/0022-1236(89)90056-6View
Published (Version of record) Open Access

Abstract

We consider the integrability problem for Lie algebras of (generally unbounded) operators in Banach space X . In addition, a Lie group G is given acting strongly continuously on X . Smoothness is defined as a relative notion with respect to the “ basepoint action .” We consider a class of smooth perturbations of Lie algebras and establish integrability for the perturbed operator Lie algebra. We also have a structure theoretic result for the components of the Levi decomposition of the perturbed Lie algebra. We give applications to automorphic Lie actions on C ∗ -algebras, and to Lie algebras of derivations. A sequel paper restricts the setting further to the case of the irrational rotation C ∗ -algebras. There a classification of smooth actions is given using the general results of the present paper.

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