  We propose an abstract framework of a kind of representation theory for
$C^*$-flows, i.e., $C^*$-algebras equipped with one-parameter automorphism
groups, as a proper generalization of Olshanski's formalism of unitary
representation theory for infinite-dimensional groups such as the
infinite-dimensional unitary group $\mathrm{U}(\infty)$. The present framework,
in particular, clarifies some overlaps and/or similarities between a certain
unitary representation theory of infinite-dimensional groups and existing works
in operator algebras, and captures arbitrary projective chains arising from
links.
