  A manifestation of the black hole information loss problem is that the
two-point function of probe operators in a large Anti-de Sitter black hole
decays in time, whereas, on the boundary CFT, it is expected to be an almost
periodic function of time. We point out that the decay of the two-point
function (clustering in time) holds important clues to the nature of observable
algebras, states, and dynamics in quantum gravity.
  We call operators that cluster in time "mixing" and explore the necessary and
sufficient conditions for mixing. The information loss problem is a special
case of the statement that in type I algebras, there exists no mixing
operators. We prove that, in a thermofield double (KMS state), if mixing
operators form an algebra (close under multiplication) the resulting algebra
must be a von Neumann type III$_1$ factor. In other words, the physically
intuitive requirement that all non-conserved operators should diffuse is so
strong that it fixes the observable algebra to be an exotic algebra called a
type III$_1$ factor. More generally, for an arbitrary out-of-equilibrium state
of a general quantum system (von Neumann algebra), we show that if the set of
operators that mix under modular flow forms an algebra it is a type III$_1$ von
Neumann factor.
  In a theory of Generalized Free Fields (GFF), we show that if the two-point
function of GFF clusters in time all operators are mixing, and the algebra is a
type III$_1$ factor. For instance, in $\mathscr{N=4}$ SYM, above the
Hawking-Page phase transition, clustering of the single trace operators implies
that the algebra is a type III$_1$ factor, settling a recent conjecture of
Leutheusser and Liu. We explicitly construct the C$^*$-algebra and von Neumann
subalgebras of GFF associated with time bands and more generally, open sets of
the bulk spacetime using the HKLL reconstruction map.
