  We formulate the baby universe construction rigorously by giving a primordial
role to the algebra of observables of quantum gravity rather than the Hilbert
space. Utilizing diffeomorphism invariance, we study baby universe creation and
annihilation via change in topology. We then construct the algebra of boundary
observables for holographic theories and show that it enhances to contain an
'extra' Abelian tensor factor to describe the bulk in the quantum regime; via
the gravitational path integral we realize this extra tensor factor, at the
level of the Hilbert space, in the context of the GNS representation. We
reformulate the necessary assumptions for the "baby universe hypothesis" using
the GNS representation. When the baby universe hypothesis is satisfied, we
demonstrate that the "miraculous cancellations" in the corresponding
gravitational path integral have a natural explanation in terms of the
character theory of Abelian $C^\ast$-algebras. We find the necessary and
sufficient mathematical condition for the baby universe hypothesis to hold, and
transcribe it into sufficient physical conditions. We find that they are
incompatible with a baby universe formation that is influenced by any bulk
process from the AdS/CFT correspondence. We illustrate our construction by
applying it to two settings, which leads to a re-interpretion of some
topological models of gravity, and to draw an analogy with the topological
vacua of gauge theory.
