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

Spherical representations of $C^*$-flows I