The planar algebra of a coaction

We study actions of compact quantum groups'' on finite quantum 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 \phi :A\to\c, a version of Jones’ ``basic construction’’ applies. This produces a certain \c^*-algebra structure on $A^{\otimes n}$, plus a coaction $v_n :A^{\otimes n}\to A^{\otimes n}\otimes H$,forevery$n$\mathrm{.}Theelements$x$satisfying$v_n(x)=x\otimes 1$arecalledfixedpointsof$v_n$\mathrm{.}Theyforma$\c^$-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.