Recently, Jensen [3], Lazar [4] and Rothwell [6] have investigated in detail the structure of scattered C*-algebras which can be regarded as the most elementary generalization of the finite dimensional involutive algebras. A C*-algebra si is called scattered if every positive functional of si is the sum of a sequence of pure functionals. The object of this note is to give a short proof of a nat...

Contents Chapter 1. Martingales, continued 1 1.1. Martingales indexed by partially ordered sets 1 1.2. Notions of convergence for martingales 3 1.3. Uniform integrability 4 1.4. Convergence of martingales with directed index sets 6 1.5. Application: The 0-1 law of Kolmogorov 8 1.6. Continuous-time martingales 9 1.7. Tail bounds for martingales 12 1.8. Application: The Pólya urn 13 1.9. Applicat...

We define a class of higher-order linear maps that transform quantum operations into quantum operations and satisfy suitable requirements of normality and complete positivity. For this class of maps we prove two dilation theorems that are the analogues of the Stinespring and Radon-Nikodym theorems for quantum operations. A structure theorem for probability measures with values in this class of ...