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 all pure state transformations of general C*-algebras that appear as restrictions of duals of linear operators on the underlying algebras. This result may variously be interpreted as either a non-commutative Banach-Stone theorem, or (in the bijective case) a pure state-based description of Wigner symmetries. These results extend the work of Shultz [Comm. Math. Phys. 82 (1982) 497-509] (who considered only the case of bijections), and also complements and completes the investigation of linear maps with pure state preserving adjoints begun in [Labuschagne and Mascioni, Adv. Math. 138 (1998) 15-45].