Title:

Topics in noncommutative probability theory with applications to statistical mechanics

Chapter I contains a presentation of NonCommutative Integration theory. The relation Between Segal's and Nelson's definition of measurability is investigated, and a new proof of duality for noncommutative probability L_{p} spaces is given. In chapter II, known results on isometries between Banach spaces of functions and operators are presented, and a new proof of the fact that unitpreserving isometries of abelian C* algebras are *isomorphisms is given. It is shown that unitpreserving *isometries between noncommutative probability L_{ p} spaces come from Jordan *homomorphisms and several conclusions are drawn. Chapter III is a presentation of TomitaTakesaki theory. Possible generalizations are pointed out, and the RadonNikodym theorem is discussed. In chapter IV the characterization of equilibrium in Quantum Statistical Mechanics by the KMS condition is investigated. In chapter V, a class of Gibbs states w_{beta} is defined on the algebra M of the canonical commutation relations in infinitely many degrees of freedom. This is done by showing that for any beta > 0 the second quantization H of a hamiltonian with positive polynomially bounded discrete spectrum defines a nuclear operator exp(betaH) from Fock space into g, a generalization of Schwartz space for infinitely many variables. This allows the construction of an "almost modular" Hiblert subalgebra on which the modular automorphisms may he defined, and satisfy the KMS condition. The final chapter contains a proof of a commutation theorem, namely that the commutant of in the GNS representation induced by wg is invariant under the modular automorphisms, and is isomorphic to its own commutant via an antiunitary involution of the GNS Hilbert space. This is done by showing that pi_{ beta} is unitarily equivalent to left multiplication on HilbertSchmidt operators on Pock space, acting on a suitable tensor product of g with Fock space.
