CONTENTS -- Introduction -- Probability Measures in Terms of Creation, Annihilation, and Neutral Operators Luigi Accardi, Hui-Hsiung Kuo and Aurel Stan -- 1. Creation, annihilation, and neutral operators -- 2. Polynomially symmetric measures -- 3. Polynomially factorizable measures -- 4. Probability measures by means of the CAN operators -- 5. Probability measures on the real line -- 5.1. Probability measures on R with finite support -- 5.2. Probability measums on R with infinite support -- 5.3. Probability measures on R with compact support -- 6. Classical probability measures on the real line -- Acknowledgments -- References -- Semi Groupes Associes a l'operateur de Laplace-Levy Luigi Accardi and Habib Ouerdiane -- 1. Introduction et prkliminaires -- 1.1. Fonctions differentiables -- 1.2. Fonctions entieres a croissance exponentielle -- 1.3. Transformation de Laplace -- 1.4. Reprdsentation intdgmle de distributions positives -- 2. Laplacien de LQvy agissant sur les distributions -- 3. Semi groupe de LQvy agissant sur l'espace des fonctions -- References -- Stochastic Golden Rule for a System Interacting with a Fermi Field Luigi Accardi, R.A. Roschin and I.V. Volovich -- 1. Introduction -- 2. Hamiltonian of the model -- 3. The "Commutative" case -- 4. Proof of Theorem 2. -- 4.1. Quasi-commutation mles for bt and Ut. -- 4.2. Normally ordered equation for Ut . -- 4.3. Inner Langevin equation. -- 4.4. Canonical form of the Jangevin equation. -- 4.5. The Master equation. -- 5. "Anti-commutative" case -- 6. Conclusions -- References -- Generating Function Method for Orthogonal Polynomials and Jacobi-Szego Parameters Nobuhiro Asai, Izumi Kubo and Hui-Hsiung Kuo -- 1. Accardi-Boiejko unitary isomorphism -- 2. Pre-generating and generating functions -- 3. Computation of orthogonal polynomials and Jacobi-Szegii parameters.

4. Orthogonal polynomials in terms of differential or difference operators -- 5. Classical examples -- Acknowledgments -- References -- Low Temperature Superconductivity and the Stochastic Limit Fabio Bagarello -- 1. Introduction -- 2. The physicals model and its stochastic limit -- 3. Possible generalizations -- References -- Multiquantum Markov Semigroups, Interacting Branching Processes and Nonlinear Kinetic Equations. Finite Dimensional CaseV.P. Belavkin and C.R. Williams -- 1. Introduction -- 2. The algebraic theory of finite dimensional multi-particle sys tems -- 3. Transitional transformations -- 4. Markovian multi-particle processes -- 5. Interaction free branching processes -- 6. Branching systems with interaction strength -- References -- A Note on Vacuum- Adapted Semimartingales and Monotone Independence Alexander C. R. Belton -- 1. Introduction -- 1.1. Conventions and Notation -- 2. Preliminaries -- 3. Result -- 4. Example -- Acknowledgments -- References -- Quantum Stochastic Processes and Applications Mohamed Ben Chrouda, Mohamed El Oued and Habib Ouerdiane -- 1. Introduction and preliminaries -- 2. Characterization theorems -- 2.1. The Operator symbol -- 2.2. Convergence of operators -- 3. The Convolution product of operators -- 3.1. Application to quantum stochastic diflerential equations -- References -- Regular Quantum Stochastic Cocycles have Exponential Product Systems B.V. Rajarama Bhat and J. Martin Lindsay -- 1. Product systems of Hilbert W*-modules -- 2. Quantum stochastic cocycles -- 3. Product system of a stochastic cocycle -- 4. Irreducible quantum stochastic dilations -- Acknowledgements -- References -- Evolution of the Atom-Field System in Interacting Fock SpaceP. K. Das -- 1. Introduction -- 2. Definitions and Preliminaries -- 3. Interaction of a single two-level atom with a single mode field.

4. Evolution of the atom-field system -- 5. Conclusion -- References -- Quantum Mechanics on the Circle through Hopf q-Deformations of the Kinematical Algebra with Possible Applications to LQvy Processes V.K. Dobrev, H.-D. Doebner and R. Twarock -- 1. Introduction -- 2. Short review of Borel quantisation with application to S1 -- 2.1. The kinematics -- 2.2. The dynamics -- 2.3. The kinematical algebra K(S1) and a family of evolution equations -- 3. A discrete quantum kinematics based on a Hopf q-deformation of the kinematical algebra -- 3.1. A Hopf q-deformation of the kinematical algebra -- 3.1.1. The position observables -- 3.1.2. The momentum observables -- 3.1.3. Coupling between the subalgebras: the Hopf q-deformation of the kinematical algebra -- 3 . 2 . The quantum kinematics on S1 -- 4. The quantum dynamics with Hopf q-structure -- 4.1. Derivation of a nonlinear q-Schrodinger equation -- 4.1.1. The right hand side of (37) -- 4.1.2. The left hand side of (37) -- 4.2. The q-deformed evolution equation -- 4.3. The continuous limit -- 4.4. Comparison with previous results -- 5. Applications to LQvy processes -- Acknowledgments -- References -- On Algebraic and Quantum Random Walks Demosthenes Ellinas -- 1. Introduction -- 2. Algebraic Random Walks -- 2.1. The case of R -- 2.2. The case of ZN -- 2.3. The case of hw-Algebra -- 3. Quantum Random Walks -- 3.1. The case of 2 -- 4. Relation of Algebraic and Quantum Random Walks -- 5. Discussion -- 6. Acknowledgments -- References -- Dual Representations for the Schrodinger Algebra Philip Feinsilver and Rent! Schott -- 1. Introduction -- 2. Dual representations -- 2.1. Dual representations and r-matrices -- 2.2. Commutation relations: Kirillov matriz, adjoint representation -- 2.3. Lie flags -- 2.4. Dual representations and flags -- 2.5. Dual representations for the Schrodinger algebra.

2.6. Double dual -- 3. Splitting formula for the SchrSdinger group -- 3.1. Splitting lemma -- 3.2. Coodinates of the second kind found -- 3.3. Basic polynomials and quantum observables -- References -- Harmonic Analysis on Non- Amenable Coxeter Groups Gero Fendler -- 1. Introduction -- 2. Coxeter groups -- 3. Geometric Representation -- 4. Adjoint Representation -- 5. A Product of Trees -- 6. Weak Amenability -- 7. Haagerup Inequality -- 8. Simplicity of the regular C*-algebra -- References -- A Limit Theorem for Conditionally Independent Beam Splittings K.H. Fichtner, Volkmar Liebscher and Masanori Ohya -- 1. Introduction -- 2. Basics -- 3. The Results -- 4. Proofs -- References -- On Factors Associated with Quantum Markov States Corresponding to Nearest Neighbor Models on a Cayley Tree Francesco Fidaleo and Farruh Mukhamedov -- 1. Introduction -- 2. Definitions and preliminary results -- 3. Construction of Gibbs states for the A-model -- 4. Diagonal states and corresponding von Neumann algebras -- 5. Applications and examples -- 5.1. Potts model -- 5.2. Markov random fields -- Acknowledgements -- References -- On Quantum Logical Gates on a General Fock Space Wolfgang Freudenberg, Masanori Ohya and Noburo Watanabe -- 1. Introduction -- 2. The General Model -- 3. The Boson Fock Space -- 4. The Quantum Channel -- 5. Control Gate -- 6. Final Output AEter Second Beam Splitting -- References -- The Chaotic Chameleon Richard D. Gill -- 1. Introduction -- 2. The Problem -- 3. The Solutions -- 4. Variant 1: Coincidences -- 5. Variant 2: Demanding More -- Acknowledgments -- References -- On an Argument of David Deutsch Richard D. Gill -- 1. Introduction: are quantum probabilities fixed by quantum determinism? -- 2. Assumptions: degeneracy, functional invariance, unitary invariance -- 3. The first part of the proof -- 4. Completing the proof -- 5. Discussion.

Acknowledgments -- References -- Volterra Representations of Gaussian Processes with an Infinite-Dimensional Orthogonal Complement Yuji Hibino and Hiroshi Muraoka -- 1. Introduction -- 2. Stationary processes -- 3. Volterra representations -- 4. Concluding remark -- References -- The Method of Double Product Integrals in Quantisation of Lie Bialgebras Robin L. Hudson -- 1. Introduction. -- 2. The It6 Hopf algebra. -- 3. Calculus in the It6 Hopf algebra. -- 4. Simple product integrals with system algebra. -- 5. Double product integrals. -- 6. The quantum Yang-Baxter equation. -- References -- On Noncommutative Independence Romuald Lenczewski -- 1. Introduction -- 2. Boolean Independence -- 3. Monotone Independence -- 4. F'reeness -- 5. Mixed types of independence -- References -- Lkvy Processes and Jacobi Fields Eugene Lytvynov -- 1. Introduction -- 2. Jacobi Matrix and its Spectral Measure -- 3. Chaos Expansion for Gaussian and Poisson Process -- 4. Jacobi field of a LQvy process -- 5. Process of Meixner's type -- 6. The usual Fock space representation of the Jacobi field of a LQvy process -- Acknowledgments -- References -- Ic-Decomposability of Positive Maps Wtadystaw A. Majewski and Marcin Marciniak -- 1. Introduction -- 2. Tomita-Takesaki scheme for transposition -- 3. k-decomposability at the Hilbert-space level -- 4. Application of local decomposability to low dimensional cases -- References -- An Introduction to LBvy Processes in Lie Groups Ming Liao -- 1. Some general theory -- 2. LQvy processes in compact Lie groups -- 3. Invariant Markov processes in homogeneous spaces -- 4. Limiting properties of LQvy processes -- 5. Dynamical aspect -- References -- Duality of W*-Correspondences and Applications Paul S. Muhly and Baruch Solel -- 1. Introduction -- 2. Correspondences and operator algebras -- 3. Duality of W*-correspondences.

4. Applications of duality: Commutants and Canonical models.