Hamiltonians, derivations and operator algebras

Abstract

This thesis falls into two parts. In Part A, Tomita-Takesaki theory is extended to the unbounded CCR-algebra A in infinitely many degrees of freedom. A particular Hamiltonian with polynomially bounded spectrum defines the Gibbs state obeta on A. It is shown that A admits a modular operator and that obeta is a KMS-state with respect to the modular automorphism. In the GNS-representation induced by obeta, the commutant pibeta(A)' is shown to satisfy the conclusion of Tomita's theorem. This is done by constructing another representation of A - on the Hilbert-Schmidt operators - for which Tomita's result is known. In Part B, perturbations of dynamical systems are considered. For a C*-dynamical system (A,alpha) with generator delta and AcB(H), delta is perturbed by a derivation Delta on A and it is shown that delta+[upper case delta] generates an automorphism group of A if Delta is inner, and an automorphism group of A" if Delta is polynomially relatively bounded. Finally, a result by Buchholz and Roberts on bounded perturbations is generalised. For two W*-dynamical systems (M,alpha), (M, beta) with generators deltaalpha and deltabeta, respectively, it is shown that, under a local commutativity condition on alpha and beta, the norm-proximity of alpha and beta on [equation] is described in terms of the operator on [equation], where gamma is a linear operator mapping D(deltaalpha) into D(deltabeta).