Pure state transformations induced by linear operators

We generalise Wigner's theorem to its most general form possible for B(h) in the sense of completely characterising those vector state transformations of B(h) that appear as restrictions of duals of linear operators on B(h). We then use this result to similarly characterise the pure state transformations of general C*-algebras that appear as restrictions of duals of linear operators on the underlying algebras. This result may be interpreted as a noncommutative Banach-Stone theorem.Comment: 24 pages, amslatex, revised and debugged versio

A crossed product approach to Orlicz spaces

We show how the known theory of noncommutative Orlicz spaces for semifinite von Neumann algebras equipped with an fns trace, may be recovered using crossed product techniques. Then using this as a template, we construct analogues of such spaces for type III algebras. The constructed spaces naturally dovetail with and closely mimic the behaviour of Haagerup $L^p$-spaces. We then define a modified $K$-method of interpolation which seems to better fit the present context, and give a formal prescription for using this method to define what may be regarded as type III Riesz-Fischer spaces.Comment: 39 pages, typos removed, presentation streamlined, non-essential results remove

Maximal Ergodic Inequalities for Banach Function Spaces

We analyse the Transfer Principle, which is used to generate weak type maximal inequalities for ergodic operators, and extend it to the general case of $\sigma$-compact locally compact Hausdorff groups acting measure-preservingly on $\sigma$-finite measure spaces. We show how the techniques developed here generate various weak type maximal inequalities on different Banach function spaces, and how the properties of these function spaces influence the weak type inequalities that can be obtained. Finally, we demonstrate how the techniques developed imply almost sure pointwise convergence of a wide class of ergodic averages.Comment: 46 pages. The former Lemma 4.7 and Theorem 4.8 (which had a small gap in the proof) is replaced by Theorem 4.7. This change affects the latter part of section

A Helson-Szeg\"o theorem for subdiagonal subalgebras with applications to Toeplitz operators

We formulate and establish a noncommutative version of the well known Helson-Szego theorem about the angle between past and future for subdiagonal subalgebras. We then proceed to use this theorem to characterise the symbols of invertible Toeplitz operators on the noncommutative Hardy spaces associated to subdiagonal subalgebras