This volume deals with structure theorems and models for finite and countable families of self-adjoint operators which satisfy commutative, noncommutative, Lie and more general relations. Such families are studied because of their relation to mathematical models of quantum systems in theoretical physics, representation theory of Lie groups and Lie algebras, (in particular, 'large' Lie algebras), and representation of infinitesimal objects of these supergroups and superalgebras. The methods employed are those of commutative and noncommutative probability theory.
For mathematicians and physicists whose work involves spectral theory, Lie algebras and probability theory. The book will also be of use to those interested in mathematical models of quantum systems.
Table of ContentsComments to the introduction.- I Families of Commuting Normal Operators.- 1. Spectral Analysis of Countable Families of Commuting Self-Adjoint Operators (CSO).- 2. Unitary Representations of Inductive Limits of Commutative Locally Compact Groups.- 3. Differential Operators With Constant Coefficients In Spaces of Functions of Infinitely Many Variables.- Inductive Limits of Finite-Dimensional Lie Algebras and Their Representations.- 4. Canonical Commutation Relations (CCR) of Systems with Countable Degrees of Freedom.- 5. Unitary Representations of The Group of Finite SU(2)-Currents on A Countable Set.- 6. Representations of The Group of Upper Triangular Matrices.- 7. A Class of Inductive Limits of Groups and Their Representations.- Collections of Unbounded Self-Adjoint operators Satisfying General Relations.- 8. Anticommuting Self-Adjoint Operators.- 9. Finite and Countable Collections of Gradedcommuting Self-Adjoint Operators (GCSO).- 10. Collections Of Unbounded CSO (Ak) And CSO (Bk) Satisfying General Commutation Relations.- Representations of Operator Algebras And Non-Commutative Random Sequences.- 11. C* -ALGEBRASU0? And Their Representations.- 12. Non-Commutative Random Sequences and Methods for Their Construction.