GROUP-ACTIONS ON C-ASTERISK-ALGEBRAS, 3-COCYCLES AND QUANTUM-FIELD THEORY

We study group extensions Delta --> Gamma --> Omega, where Gamma acts on a C-algebra A. Given a twisted covariant representation pi, V Of t he pair A, Delta we construct 3-cocycles on Omega with values in the c entre of the group generated by V(Delta). These 3-cocycles are obstruc tions to the existence of an extension of Omega by V(Delta) which acts on A compatibly with Gamma. The main theorems of the paper introduce a subsidiary invariant Lambda which classifies actions of Gamma on V(D elta) and in terms of which a necessary and sufficient condition for t he the cohomology class of the 3-cocycle to be non-trivial may be form ulated. Examples are provided which show how non-trivial 3-cocycles ma y be realised, The framework we choose to exhibit these essentially ma thematical results is influenced by anomalous gauge field theories. We show how to interpret our results in that setting in two ways, one mo tivated by an algebraic approach to constrained dynamics and the other by the descent equation approach to constructing cocycles on gauge gr oups. In order to make comparisons with the usual approach to cohomolo gy in gauge theory we conclude with a Lie algebra version of the invar iant Lambda and the 3-cocycle.