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Coactions of compact groups on $M_n$

S. Kaliszewski, Magnus B. Landstad, John Quigg

Abstract

We prove that every coaction of a compact group on a finite-dimensional $C^*$-algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on $M_n$ is inner if and only if its fixed-point algebra has an abelian $C^*$-subalgebra of dimension $n$. Investigating the existence of effective ergodic coactions on $M_n$ reveals that $\operatorname{SO}(3)$ has them, while $\operatorname{SU}(2)$ does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on $M_n$.

Coactions of compact groups on $M_n$

Abstract

We prove that every coaction of a compact group on a finite-dimensional -algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on is inner if and only if its fixed-point algebra has an abelian -subalgebra of dimension . Investigating the existence of effective ergodic coactions on reveals that has them, while does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on .

Paper Structure

This paper contains 8 sections, 9 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fell bundle s
  4. Unitary implementation
  5. Inner
  6. Ergodic
  7. $G=S_3$
  8. Conclusion

Key Result

Theorem 3.1

Let $\delta$ be a coaction of a compact group $G$ on a finite-dimensional $C^*$-algebra $A$. Then there is a Fell bundle $\mathcal{A}$ over $G$ such that $(A,\delta)$ is isomorphic to the dual coaction $(C^*(\mathcal{A}),\delta_\mathcal{A})$.

Theorems & Definitions (24)

  • Theorem 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 5.1
  • proof
  • ...and 14 more