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Title:  Groups of automorphisms of operator algebras 
Authors:  Moffat, James 
Issue Date:  1974 
Publisher:  Newcastle University 
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 Ginvariant states and the ergodic action of G on subalgebras
of R (G acts ergodically if 0 and I are the only invariant
projections). 
Description:  PhD Thesis 
URI:  http://hdl.handle.net/10443/1646 
Appears in Collections:  School of Mathematics and Statistics

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