Groups of automorphisms of operator algebras

Abstract

An important part of the theory of locally compact groups is the study of their unitary representations. In this thesis, we study the representation of such groups, and more general topological groups, as groups of automorphisms of C*-algebras. Certain types of continuity and measurability of such representations (which we call automorphic representations) are defined and shown to be equivalent in certain cases. We consider a continuous representation, α, of an abelian connected topological group G as a group of automorphisms of a C*-algebra U acting on a Hilbert space H. The topology on α(G) is that derived as a subset of the Banach space of bounded operators on U. Such a representation is shown to be equivalent to a norm continuous unitary representation g→ Ug of G by unitaries Ug in the weak operator closure of U, such that α(g)(A) = Ug A U*g (g Є G, A Є U). In the case of a locally compact group G and a weaker continuity condition on the representation α, we obtain (when U is a factor or a separable simple C*-algebra with unit) a necessary and sufficient condition that there exist a strongly continuous unitary representation g→ Ug of G by unitaries Ug Є U such that α(g)(A) = UgAU*g (A Є U, 9 Є G). If G is a group of automorphisms of a von Neumann algebra an equivalence relation can be defined, in terms of G, on the projections in R, which extends the usual definition of equivalence of projections. We show that certain results concerning the type of the tensor product of von Neumann algebras carry over to this more general situation. Ergodic theory is essentially the study of groups of transformations of a measure space (X, μ). If X is a locally compact space, L∞(X, μ) is an abelian von Neumann algebra. We prove that certain results concerning the existence of an equivalent measure on X invariant under the transformation group carry over to the case of an amenable group G of automorphisms of a general von Neufnann algebra R. This, gives a necessary and sufficient condition for the existence of a faithful normal state on R invariant under G. We also show that a link exists between normal extremal G-invariant states and the ergodic action of G on subalgebras of R (G acts ergodically if 0 and I are the only invariant projections).