Contents of

Stochasticity and Partial Order
P.M. Alberti and A. Uhlmann
VEB Deutscher Verlag der Wissenschaften
Berlin 1981

Introduction , 7

1. Some classical results 9

1.1. Elementary notations 9
1.2. The linear space \ell^1 in n dimensions 9
1.3. Stochastic and doubly stochastic maps 11
1.4. The relation \succ, 16
1.5. An example: Comparison of Gibbsian states .. 19
1.6. An example: Localization of spectra for Hermitian matrices . 20
1.7. Convexity and the relation \succ 23
1.8. Examples: Some maps and processes 28
1.9. A partial order of m-tuples a_1,...,a_m 31

2. Order structures of matrices 37

2.1. The relation \succ in the space B^1 37
2.2. Some doubly stochastic maps, 40
2.3. Convexity and therelation \succ 45
2.4. Another partial order.. 47
2.5. Further examples: A^k \rhd |A|^k and related inequalities 52
2.6. Miscellaneous 55

3. The order structure in the state space of C*- and W*-algebras 58

3.1. Preliminaries 58
3.2. The relation \succ; definitions and elementary results.. 60
3.3. Some essential results for W*-algebras 63

4. The c-ideal 65

4.1. Definition and the existence of the c-ideal  65
4.2. Ky Fan functionals and von Neumann's relation  68
4.3. A centre-valued convex trace, 71
4.4. Maximally mixed states (chaotic states)  76
4.5. Technical Lemmata ... 78

5. The \Sigma-property . 81

5.1. The definition, preliminary remarks 81
5.2. The case of properly infinite projections - type III 82
5.3. The finite case . 83
5.4. The universality of the \Sigma-property for W*-algebras 85
5.5. A duality theorem, remarks 87
5.6. Positive linear forms on C*-algebras and the \Sigma-property 87

6. The dual structure in W*-algebras 91

6.1. Introduction 91
6.2. Technical preliminaries.92
6.3. The purely infinite case 94
6.4. Approximation results, 97
6.5. A characterization of (M_+,\succ) 105
6.6. The main theorem 107
6.7. A special case: Finite W*-algebras III
6.8. Examples for \succ on M_+ 113

References 116

Symbols 121

Keywords 122