Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 119-158.

The planar algebra of a coaction

Authors: T. Banica
Author institution: D\' epartement de Math\' ematiques, Universit\' e Paul\break Sabatier, 118 route de Narbonne, 31062 Toulouse, France

Summary: We study actions of {\em compact quantum groups} on {\em finite quantum} \break {\em spaces}. According to Woronowicz and to general

$C^*$ -algebra philosophy, these correspond to certain coactions

$v:A\to A\otimes H$ . Here

$A$ is a finite dimensional

$C^*$ -algebra, and

$H$ is a certain special type of Hopf

$*$ -algebra. If

$v$ preserves a positive linear form

$\varphi :A\to\complex$ , a version of Jones' {\em basic construction} applies. This produces a certain

$C^*$ -algebra structure on

$A^{\otimes n}$ , plus a coaction

$v_n :A^{\otimes n}\to A^{\otimes n}\otimes H$ , for every

$n$ . The elements

$x$ satisfying

$v_n(x)=x\otimes 1$ are called fixed points of

$v_n$ . They form a

$C^*$ -algebra

$Q_n(v)$ . We prove that under suitable assumptions on

$v$ the graded union of the algebras

$Q_n(v)$ is a spherical

$C^*$ -planar algebra.

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