Free quasi-free states.

摘要

To a real Hilbert space {dollar}{lcub}cal H{rcub}sb{lcub}IR{rcub}{dollar} with a given one-parameter group of isometries {dollar}Usb{lcub}t{rcub}{dollar} we associate a {dollar}Csp{lcub}*{rcub}{dollar}-algebra {dollar}Gamma({lcub}cal H{rcub}sb{lcub}IR{rcub},Usb{lcub}t{rcub}){dollar} with a state {dollar}phisb{lcub}U{rcub}.{dollar} The algebra is a free-probability analog of the CAR and CCR algebras, and the free quasi-free state {dollar}phisb{lcub}U{rcub}{dollar} is an analog of the quasi-free states of CAR and CCR. These {dollar}Csp{lcub}*{rcub}{dollar}-algebras are in many cases simple. {dollar}Gamma{dollar} is a functor from pairs {dollar}({lcub}cal H{rcub}sb{lcub}IR{rcub},Usb{lcub}t{rcub}){dollar} and equivariant contractions to {dollar}Csp{lcub}*{rcub}{dollar}-algebras and completely-positive maps; moreover, {dollar}Gamma{dollar} transfers direct sums into reduced free products of {dollar}Csp{lcub}*{rcub}{dollar}-algebras.; The von Neumann algebras {dollar}Gamma({lcub}cal H{rcub}sb{lcub}IR{rcub},Usb{lcub}t{rcub})sp{lcub}primeprime{rcub}{dollar} arising in the GNS representation for {dollar}phisb{lcub}U{rcub}{dollar} are free analogs of the Araki-Woods factors. These algebras are factors if dim {dollar}{lcub}cal H{rcub}sb{lcub}IR{rcub}>1,{dollar} and of type III if {dollar}Usb{lcub}t{rcub}{dollar} is non-trivial. We compute Connes' {dollar}tau{dollar} invariant, and show that it is a complete invariant for the free Araki-Woods factors coming from non-trivial almost-periodic one-parameter groups {dollar}Usb{lcub}t{rcub}.{dollar} We also give a description of the cores of the free Araki-Woods factors using some constructions from algebra-valued free probability theory.