Relative modular theory for a weight

Abstract

We consider the balanced weight χ \chi of a semi-finite weight φ \varphi and a (not necessarily faithful) normal positive functional ψ \psi on a von Neumann algebra M \mathcal M , and discuss how the modular operator Δ χ \Delta _\chi and the modular conjugation J χ J_\chi are described under the identification of the standard Hilbert space H χ \mathcal {H}_\chi with H φ ⊕ p H φ ⊕ p ′ H φ ⊕ p p ′ H φ \mathcal {H}_\varphi \oplus p\mathcal {H}_\varphi \oplus p’\mathcal {H}_\varphi \oplus pp’\mathcal {H}_\varphi , where p p is the support projection of ψ \psi and p ′ = J φ p J φ ∈ M ′ p’=J_\varphi p J_\varphi \in \mathcal {M}’ .