A Variational Expression for the Relative Entropy

We prove that for the relative entropy of faithful normal states φ and ω on the von Neumann algebra M the formula S(φ,ω) = sup{ω(Λ)-logφ(/): h = h*eM] holds. In general von Neumann algebras the relative entropy was defined and investigated by Araki [1, 3]. After Lieb had proved the joint convexity of the relative entropy in the type / case [10] several proofs appeared in the literature and they all benefited from the operator convexity of the function t~>— logί [8, 11]. Improving a result of Pusz and Woronowicz [14] Kosaki [9] obtained a variational formula for the relative entropy, which allows to extend the notion also to C*-algebras. The expression we are going to deal with is of a different kind. It shows that the relative entropy S(φ, ω) as a function of φ is the conjugate convex function (i.e., Legendre transform) of the convex function /z->logφ(/), where φ denotes the inner perturbation of the state φ by the selfadjoint operator h. The perturbed state φ was used by Araki to extend the Golden-Thompson inequality ([7, 18], see also [15]) to traceless von Neumann algebras. Approaching our variational expression for the relative entropy we generalize the GoldenThompson-Araki inequality [2] essentially and we state also the exact condition for the equality. If φ and ω are faithful normal states of the von Neumann algebra M then the relative entropy is defined by means of the relative modular operator Δ(φ, ω). If Ω is the vector representative of ω in the natural positive cone P then S(φ, ω) . The variational expression of Kosaki says that