CONTENTS -- CONTENTS OF QPC XI -- PREFACE for QPC Volumes XI & XII -- INTEGRAL-SUM KERNEL OPERATORS -- 0. INTRODUCTION -- 1.1. Some products on F(Γ). -- 1.2. Product functions. -- 1.3. Guichardet Space. -- 1. FINITE POWER SETS -- 2. INTEGRAL-SUM CONVOLUTIONS -- 2.1. Duality Transforms. -- 2.2. Formal Derivation. -- 2.3. Basic Estimate. -- 3. QUANTUM WIENER INTEGRALS -- 4. INTEGRAL-SUM KERNEL OPERATORS -- 4.1. Basic Estimate. -- 4.2. Uniqueness of the kernel. -- 4.3. Reconstruction of kernel from operator. -- 4.4. Algebras of integral-sum kernel operators. -- 4.5. Four argument integral-sum kernels. -- 4.6. Matrix-valued kernels. -- CONCLUSION -- BIBLIOGRAPHICAL NOTES -- REFERENCES -- QUANTUM PROBABILITY APPLIED TO THE DAMPED HARMONIC OSCILLATOR -- 1. THE FRAMEWORK OF QUANTUM PROBABILITY -- 2. SOME QUANTUM MECHANICS -- 3. CONDITIONAL EXPECTATIONS AND OPERATIONS -- 4. SECOND QUANTISATION -- 5. UNITARY DILATIONS OF SPIRALING MOTION -- 6. THE DAMPED HARMONIC OSCILLATOR -- REFERENCES -- QUANTUM PROBABILITY AND STRONG QUANTUM MARKOV PROCESSES -- 0. INTRODUCTION -- I. Quantum Probability -- 1. A COMPARATIVE DESCRIPTION OF CLASSICAL AND QUANTUM PROBABILITY -- 2. THE ROLE OF TENSOR PRODUCTS OF HILBERT SPACES -- 3. SOME BASIC OPERATORS ON FOCK SPACES -- 4. FROM URN MODEL TO CANONICAL COMMUTATION RELATIONS -- II. Quantum Markov Processes -- 5. STOCHASTIC OPERATORS ON C*-ALGEBRAS -- 6. STINESPRING'S THEOREM -- 7. EXTREME POINTS OF THE CONVEX SET OF STOCHASTIC OPERATORS -- 8. STINESPRING'S THEOREM IN TWO STEPS -- 9. CONSTRUCTION OF A QUANTUM MARKOV PROCESS -- 10. THE CENTRAL PART OF MINIMAL DILATION -- 11. ONE PARAMETER SEMIGROUPS OF STOCHASTIC MAPS ON A C*-ALGEBRA -- III. Strong Markov Processes -- 12. NONCOMMUTATIVE STOP TIMES -- 13. MARKOV PROCESS AT SIMPLE STOP TIMES -- 14. MINIMAL MARKOV FLOW AT SIMPLE STOP TIMES.

15. STRONG MARKOV PROPERTY OF THE MINIMAL FLOW FOR A GENERAL STOP TIME -- 16. STRONG MARKOV PROPERTY UNDER A SMOOTHNESS CONDITION -- 17. A QUANTUM VERSION OF DYNKIN'S LOCALIZATION FORMULA -- ACKNOWLEDGEMENTS -- REFERENCES -- LIMIT PROBLEMS FOR QUANTUM DYNAMICAL SEMIGROUPS - INSPIRED BY SCATTERING THEORY -- 0. INTRODUCTION -- 1. COMPARISON OF THE LARGE TIME BEHAVIOUR OF TWO SEMIGROUPS -- 2. THE CLASSIFICATION OF STATES -- 3. ERGODIC PROPERTIES OF QUANTUM DYNAMICAL SEMIGROUPS -- 4. CONVERGENCE TOWARDS THE EQUILIBRIUM -- ACKNOWLEDGEMENT -- REFERENCES -- A SURVEY OF OPERATOR ALGEBRAS -- 0. COMPLEX BANACH ALGEBRAS -- 1. C*-ALGEBRAS -- 1.1. Definition and first spectral properties. -- 1.2. Adding a unit. -- 1.3. First examples: abelian C*-aIgebras. -- 1.4. Continuous functional calculus in C*-algebras. -- 1.5. More examples: B(H) and its sub-C*-algebras. -- 1.6. Order Structure, states, and t h e GNS construction. -- 1.6.1. Positive elements and order in A. -- 1.6.2. Dual order structure and states. -- 1.6.3. GNS construction. -- 2. VON NEUMANN ALGEBRAS -- 2.1. Some topologies on B(H). -- 2.1.1. Three natural topologies. -- 2.1.2. The ideal L1(H) -- 2.2. von Neuman algebras. -- 2.2.1. von Neumann bicommutant theorem. -- 2.2.2. Definition of von Neumann algebras. -- 2.2.3. Predual of a von Neumann algebra. -- 2.3. Examples. -- 2.4. Theory of representations. -- 2.4.1. The GNS section. -- 2.4.2. Examples of representations. -- 2.5. Functional calculus revisited. -- 2.5.1. Complex measures. -- 2.5.2. Spectral measures. -- 2.5.3. Borel functional calculus in von Neumann algebras. -- 3. MODULAR THEORY -- 4. CONDITIONAL EXPECTATIONS -- REFERENCES -- QUANTUM STOP TIMES -- 0. INTRODUCTION -- 1. NOTATIONS -- 2. THEORY OF INTEGRATION WITH STOP TIMES -- 3. STRONG MARKOV PROPERTY -- 4. STOPPED PROCESSES -- 5. PRE-S AND POST-S ALGEBRAS -- 6. REMARKS -- REFERENCES.

FREE CALCULUS -- 0. BASIC DEFINITIONS AND FACTS -- 1. COMBINATORIAL ASPECTS OF FREENESS: THE CONCEPT OF CUMULANTS -- 1.1. Definitions. -- 1.2. Examples. -- 1.3. Remarks. -- 1.4. Definition. -- 1.5. Remarks and Examples. -- 1.6. Theorem. -- 1.7. Remarks. -- 1.8. Notation. -- 1.9. Proposition. -- 1.10. Proposition. -- 1.11. Theorem. -- 1.12. Corollary. -- 1.13. Remark. -- 1.14. Notation. -- 1.15. Proposition. -- 1.16. Definition. -- 1.17. Theorem. -- 2. FREE STOCHASTIC CALCULUS -- 2.1. Definition. -- 2.2. Theorem. -- 2.3. Remark. -- 2.4. Remarks. -- 2.5. Remarks. -- 2.6. Definition. -- 2.7. Proposition (Itô isometry). -- 2.8. Notation. -- 2.9. Theorem (Burkholder-Gundy inequality). -- 2.10. Corollary. -- 2.11. Theorem (Itô formula - product form). -- 2.12. Example. -- 2.13. Theorem (Itô formula - functional form). -- REFERENCES -- CONTINUOUS KERNEL PROCESSES IN QUANTUM PROBABILITY -- 0. INTRODUCTION -- 1. DISCRETE KERNELS -- 2. A BASIC LEMMA AND HEURISTIC CONSIDERATIONS -- 3. MEASURABLE KERNELS -- 4. CALCULUS OF KERNELS -- 5. ADAPTED PROCESSES -- 6. A LINEAR DIFFERENTIAL EQUATION -- 7. THE SINGULAR COUPLING LIMIT -- ACKNOWLEDGEMENTS -- REFERENCES -- BIBLIOGRAPHY FOR QPC XI & XII.