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Spherical representations of $C^*$-flows I

Yoshimichi Ueda

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.

Spherical representations of $C^*$-flows I

Abstract

We propose an abstract framework of a kind of representation theory for -flows, i.e., -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 . 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.

Paper Structure

This paper contains 17 sections, 38 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. General setup and notations
  3. $(\alpha^t,\beta)$-spherical representations
  4. Abstract investigations
  5. Woronowicz's existence result on KMS states revisited
  6. Inductive flows
  7. Non-continuous flows and Local normality
  8. Reformulation of $(\alpha^t,\beta)$-spherical representations
  9. Review on $W^*$-inductive limits
  10. Locally bi-normal $(\alpha^t,\beta)$-spherical representations
  11. Viewpoint of Gelfand pairs
  12. Ergodic method
  13. Spectral decomposition
  14. Ergodic method by example
  15. Bratteli diagram description
  16. ...and 2 more sections

Key Result

Lemma 3.3

For an $(\alpha^t,\beta)$-spherical representation $(\Pi : A^{(2)} \curvearrowright \mathcal{H}_\Pi,\xi)$ of $A$, we define Then we have for every $a\in A_{\alpha^t}^\infty$.

Theorems & Definitions (79)

  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Definition 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • proof
  • Proposition 3.7
  • ...and 69 more