Quantum Probability for Probabilists - Paul A. Meyer

. .

Sommaire:
I: Non Commutative Probability.- II: Spin.- III: The Harmonic Oscillator.- IV: Fock Space (1).- V. Multiple Fock Spaces.- VI. Stochastic Calculus on Fock Space.- VII. Independence.- Appendix 1: Functional Analysis.- Hilbert-Schmidt operators (1).- Trace class operators (2).- Duality properties (3) Weak convergence properties (4).- Weak topologies for operators (5).- Tensor products of Hilbert spaces (6-7).- Appendix 2: Conditioning and Kernels.- Conditioning: discrete case (1).- Conditioning: continuous case (2).- Example of the canonical pair (3).- Multiplicity theory (4).- Classical kernels (5).- Non commutative kernels, first form (6).- second form (7).- Completely positive maps (8).- Some difficulties (9).- Appendix 3: Two Events.- 1. Elementary theory.- Application of spectral theory (2).- Some elementary properties (3).- Positive elements (4).- Symbolic calculus for s.a. elements (5).- Applications (6).- Characterization of positive elements (7).- A few inequalities (8).- Existence of many states (1).- Representations and the GNS theorem (2-3).- Examples from toy Fock space theory (4).- Quotient algebras and approximate units (5).- 3. Von Neumann algebras.- Weak topologies and normal states (1).- Von Neumann's bicommutant theorem (2-3).- Kaplanski's density theorem (4).- The predual (5).- Normality and order continuity (6).- About integration theory (7).- Measures with bounded density (8).- The linear Radon-Nikodym theorem (9).- The KMS condition (10).- Entire vectors (11).- 4. The Tomita-Takesaki theory.- Elementary geometric properties (1).- The main operators (2-3).- Interpretation of the adjoint (4).- The modular property (5).- Using the linear RN theorem (6).- The main computation (7).- The three main theorems (8).- Additional results (9).-Examples (10).- Appendix 5: Local Times and Fock Space.- 1. Dynkin's formula.- Symmetric Markov semigroups and processes (1).- Dynkin's formula (2).- Sketch of the Marcus-Rosen approach to the continuity of local times (3).- 2. Le Jan's supersymmetric approach.- Notations of complex Brownian motion (1).- Computing the Wiener product (2).- Stratonovich integral and trace (4).- Expectation of the exponential of an element of the second chaos (5).- Exponential formula in the antisymmetric case (7).- Supersymmetric Fock space: the Wick and Wiener products (8).- Properties of the Wiener product (9).- Applications to local times (sketch) (10).- References.- Index of Notation.