  We propose a rigorous derivation of the Bekenstein upper limit for the
entropy/information that can be contained by a physical system in a given
finite region of space with given finite energy. The starting point is the
observation that the derivation of such a bound provided by Casini [6] is
similar to the description of the black hole incremental free energy that had
been given by the first named author [23]. The approach here is different but
close in the spirit to [6]. Our bound is obtained by operator algebraic
methods, in particular Connes' bimodules, Tomita-Takesaki modular theory and
Jones' index are essential ingredients inasmuch as the von Neumann algebras in
question are typically of type III. We rely on the general mathematical
framework, recently set up in [26], concerning quantum information of infinite
systems.
