Smooth dense subalgebras and Fourier multipliers on compact quantum groups

Abstract

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 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∞D. We check explicitly that these conditions hold true on the quantum SU2q for both its 3-dimensional and 4-dimensional calculi.