Topics in non-commutative probability theory with applications to statistical mechanics

Abstract

Chapter I contains a presentation of Non-Commutative Integration theory. The relation Between Segal's and Nelson's definition of measurability is investigated, and a new proof of duality for non-commutative 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 unit-preserving isometries of abelian C* algebras are *-isomorphisms is given. It is shown that unit-preserving *-isometries between non-commutative probability L _p spaces come from Jordan *-homomorphisms and several conclusions are drawn.

Chapter III is a presentation of Tomita-Takesaki theory. Possible generalizations are pointed out, and the Radon-Nikodym 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 Hilbert-Schmidt operators on Pock space, acting on a suitable tensor product of g with Fock space.