Spherical representations of $C^*$-flows I

TL;DR

This paper introduces an abstract framework for representing $C^*$-flows, generalizing existing theories for infinite-dimensional groups and connecting them with operator algebra representations.

Contribution

It develops a new representation theory framework for $C^*$-flows, extending Olshanski's formalism to encompass broader classes of automorphisms and links.

Findings

Clarifies overlaps between infinite-dimensional group representations and operator algebras

Captures arbitrary projective chains from links

Provides a unified framework for $C^*$-flow representations

Abstract

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.