Semigroup representations : an abstract approach
Chapter One After the definitions and basic results required for the rest of the thesis, a notion of spectrum for semigroup representations is introduced and some relevant examples given. Chapter Two Any semigroup representation by isometries on a Banach space may be dilated to a group representation on a larger Banach space. A new proof of this result is presented here, and a connection is shown to exist between the dilation and the trajectories of the dual representation. The problem of dilating various types of spaces, including partially ordered spaces, C*-algebras, and reflexive spaces, is discussed, and new dilation theorems are given for dual Banach spaces and von Neumann algebras. Chapter Three In this chapter the spectrum of a representation is examined more closely with the aid of methods from Banach algebra theory. In the case where the representation is by isometries it is shown that the spectrum is non-empty, that it is compact if and only if the representation is norm-continuous, and that any isolated point in the unitary spectrum is an eigenvalue. Chapter Four An analytic characterisation is given of the spectral conditions that imply a representation by isometries is invertible. For representations of Z+n this con- dition is shown to be equivalent to polynomial convexity. Some topological conditions on the spectrum are also shown to imply invertibility. Chapter Five The ideas of the previous chapters are applied to problems of asymptotic behaviour. Asymptotic stability is described in terms of the behaviour of the dual of a representation. Finally, the case when the unitary spectrum is countable is discussed in detail.