Relative hamiltonian for faithful normal states of a von neumann algebra

Abstract

Let ψ be a cyclic and separating vector for a von Neumann algebra {M} and Δψbe its modular operator. For any elements Qi,…, Qnin {M} and complex numbers zi,…, znsuch that Re Zj≧0 and ΣRe ≦ 1/2, F is shown to be in the domain of (FOMULA OMITTED) A self adjoint operator h=h(φ/ψ)∈{M} is called a Hamiltonian of a faithful normal state φ of {M} relative to another faithful state ψ) of {M} if vectors ξφand ξψrepresenting φ and ψ (in the canonical cone (FOMULA OMITTED)) is related by (FOMULA OMITTED) The operator (FOMULA OMITTED) is shown to be an intertwining unitary operator between modular automorphisms (FOMULA OMITTED) and (FOMULA OMITTED) for states φ and ψ: (FOMULA OMITTED) The relative hamiltonian h(φ/ϕ) is unique for given states φ and ψ. It exists and satisfies (FOMULA OMITTED), where Ф1≧Ф2means that Ф1≧Ф2is in the canonical cone (FOMULA OMITTED). In particular, if l1ψ≧φl2ψ, then h(φ/ψ) exists and satisfies the above inequality. The modular operators As Δξ(φ) and Δξ(ψ) are related by where J is the common modular conjugation operator for ξφand ξψThe chain rule h(φ1/φ2)+h(φ2/φ3) = h(φ1/φ3) is satisfied. © 1973, Research Institute forMathematical Sciences. All rights reserved.