Some problems in functional analysis

The first part of this thesis is a study of strongly continuous (or weakly) continuous one parameter semigroups and groups of bounded linear maps on Banach spaces, their perturbation and scattering, paying particular attention to operator algebras.
CHAPTER 1. We develop the spectral theory of derivations and groups of automorphisms on operator algebras, and study their relationships. The spectrum of a bounded strongly continuous one parameter group on a Banach space is characterised in terms of the spectrum of its infinitesimal generator. The relationship between Arveson's and Borchers' approach to the spectral theory of groups of *‑automorphisms on operator algebras is brought out, and we consider the problem of unitary implementation.
CHAPTER 2. Following S.C. Lin's reflexive Banach space theory, we consider the perturbation, similarity and scattering of strongly continuous one parameter semigroups and groups on Banach spaces, using a time dependant approach and a strong type of smoothness, with particular reference to C*‑algebras and preduals of W*‑algebras. Some results regarding time evolution in quasi local algebras are used to derive scattering of quasi free evolution groups in the CAR algebra by inner perturbations.
CHAPTER 3. The previous chapter leads us to study scattering of ultraweakly continuous groups of σ-weakly continuous linear maps on von Neumann algebras, by smooth perturbations of a certain weak type. We give a technique for handling groups of *‑automorphisms.
CHAPTER 4. We study time dependant perturbations and scattering of strongly continuous one parameter groups on a Banach space E. The problem is raised to the higher space Lp(ℝ;E) (1 ≤ p ≤ ∞) or C0(ℝ;E), where the perturbation is made time independant and the methods of Chapter 2 apply. We characterise certain strongly continuous groups on the higher space in terms of propogators on the lower space, and show how their scattering is related. Some examples are given.
The second part of the thesis investigates the structure of completely positive maps.
CHAPTER 5. Schwartz type inequalities for n-positive linear mappings on *‑algebras are obtained. We demonstrate why the bounded completely positive linear maps for a Banach *‑algebra with approximate identity, and in particular for a C*‑algebra, should be regarded as higher order state spaces, A theorem of F. and M. Riesz is generalised to give a sufficient condition for the covariance of certain representations of a C*‑algebra relative to a one parameter group of *‑automorphisms. A completely positive analogue of Tomiyama's theorem regarding the singularity of conditional expectations on W*‑algebras is obtained. We characterise the completely positive linear mappings on the CCR algebra. Operator algebra analogues of Nagy's hilbert space dilations and Stroescu's Banach space dilations are obtained. The infinitesimal generators of strongly continuous one parameter semigroups and groups of linear maps with various positivity properties are studied.
CHAPTER 6. Unbounded completely positive linear maps or operator valued weights on C*‑algebras are defined. We construct the Stinespring representation for an unbounded completely positive linear map α. We study the natural order structure for such maps, and following van Daele for scalar valued weights, when α has dense domain, we construct a largest operator valued weight α0 majorised by α, and with the property that it is the upper envelope of continuous completely positive maps. Following Combes we study the quasi equivalence, equivalence and type of the Stinespring representation associated with operator valued weights. Following van Daele, we study an unbounded completely positive map α which is invariant under a group of *‑automorphisms and has dense domain, and construct a G-invariant projection map φ of the set ℑ of continuous completely positive maps dominated by α, onto the set ℑ0 of G invariant elements of ℑ. This is used to derive various properties of the envelope of ℑ0. Asymptotically abelian systems and their ergodic bounded completely positive maps are studied. Observations are made regarding the possible classification of the spectral type of strongly continuous one parameter groups of *‑automorphisms of a C*‑algebra which possess an invariant operator valued weight.

Advisor: Davies, E