  Bogoliubov excitations of Bose-Einstein condensates in optical lattices may
possess band topology in analogous to topological insulators in class AII of
fermions. Using the language of the Krein-space theory, this topological
property is shown to be protected by a pseudo-time-reversal symmetry that is
pseudo-antiunitary and squares to $-1$, with the associated bulk topological
invariant also being a $\mathbb Z_2$ index. We construct three equivalent
expressions for it, relating to the Pfaffian, the pseudo-time-reversal
polarization, and most practically, the Wannier center flow, all adopted from
the fermionic case, defined here with respect to the pseudo inner product. In
the presence of an additional pseudo-unitary and pseudo-Hermitian inversion
symmetry, a simpler expression is derived. We then study two toy models
feasible on cold atom platforms to numerically confirm the bulk-boundary
correspondence. The Krein-space approach developed in this work is a universal
formalism to study all kinds of symmetry-protected topological bosonic
Bogoliubov bands.
