The quantum disk is not a quantum group

In this paper, we show that the quantum disk, i.e. the quantum space correspondingto the Toeplitz C∗-algebra does not admit any compact quantum group structure. Weprove that if such a structure existed the resulting compact quantum group would simul-taneously be of Kac type and not of Kac type. The main tools used in the solution comefrom the theory of operators on Hilbert spaces.