  The AdS/CFT correspondence states that certain conformal field theories are
equivalent to string theories in a higher-dimensional anti-de Sitter space. One
aspect of the correspondence is an equivalence of density matrices or, if one
ignores normalizations, of positive operators. On the CFT side of the
correspondence, any two positive operators $A,B$ will satisfy the trace
inequality $\operatorname{Tr}(AB) \leq \operatorname{Tr}(A)
\operatorname{Tr}(B)$. This relation holds on any Hilbert space ${\cal H}$ and
is deeply associated with the fact that the algebra $B({\cal H})$ of bounded
operators on ${\cal H}$ is a type I von Neumann factor. Holographic bulk
theories must thus satisfy a corresponding condition, which we investigate
below. In particular, we argue that the Euclidean gravitational path integral
respects this inequality at all orders in the semi-classical expansion and with
arbitrary higher-derivative corrections. The argument relies on a conjectured
property of the classical gravitational action, which in particular implies a
positive action conjecture for quantum gravity wavefunctions. We prove this
conjecture for Jackiw-Teitelboim gravity and we also motivate it for more
general theories.
