The quantum disk is not a quantum group

Abstract

We show that the quantum disk, i.e. the quantum space corresponding to the Toeplitz C * -algebra does not admit any compact quantum group structure. We prove that if such a structure existed the resulting compact quantum group would simultaneously be of Kac type and not of Kac type. The main tools used in the solution come from the theory of type I locally compact quantum groups, but also from the theory of operators on Hilbert spaces.

Compact quantum group structures on type-I $\mathrm{C}^*$-algebras

“…Furthermore one can often show that certain C * -algebras do not admit a compact quantum group structure solely on the basis of some of their properties as C *algebras. Examples of such results are given in [Soł10a,Soł10b] and also [Soł14] and most recently [KS20a]. In this last paper the second and third author show that the C * -algebra known as the Toeplitz algebra (the C * -algebra generated by an isometry) does not admit a structure of a compact quantum group.…”

“…In the present paper the techniques of [KS20a] are vastly generalized and applied to a number of problems. Moreover the direct integral decompositions of representations (and other objects) are avoided.…”

Modular properties of type I locally compact quantum groups

“…In Section 5 we show how various operators act on the level of direct integrals. We remark that a formula for ∇ it ϕ from Theorem 5.4 was recently used in [8] to deduce that the Toeplitz algebra is not an algebra of continuous functions on a compact quantum group. Finally, in Section 7 we describe two interesting examples of type I locally compact quantum groups: discrete group SU q (2) and the quantum "az + b" group.…”