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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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