Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups

We define and study dense Frechet subalgebras of compact quantum groups realised as smooth domains associated with a Dirac type operator with compact resolvent. Further, we construct spectral triples on compact matrix quantum groups in terms of Clebsch–Gordon coefficients and the eigenvalues of the Dirac operator $${\mathcal{D}}$$ . Grotendieck’s theory of topological tensor products immediately yields a Schwartz kernel theorem for linear operators on compact quantum groups and allows us to introduce a natural class of pseudo-differential operators on them. It is also shown that regular pseudo-differential operators are closed under compositions. As a by-product, we develop elements of the distribution theory and corresponding Fourier analysis. We give applications of our construction to obtain sufficient conditions for Lp − Lq boundedness of coinvariant linear operators. We provide necessary and sufficient conditions for algebraic differential calculi on Hopf subalgebras of compact quantum groups to extend to our proposed smooth subalgebra $${{C}^\infty_\mathcal {D}}$$ . We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.