Smooth dense subalgebras and Fourier multipliers on compact quantum groups

Abstract

We define and study dense Frechet subalgebras of compact quantum groups consisting of elements rapidly decreasing with respect to an unbounded sequence of real numbers. Further, this sequence can be viewed as the eigenvalues of a Dirac-like operator and we characterize the boundedness of its commutators in terms of the eigenvalues. 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 compact quantum groups. 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 $L^p-L^q$ 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 the proposed smooth structure. We check explicitly that these conditions hold true on the quantum $SU(2)$ for both its 3-dimensional and 4-dimensional calculi.

arXiv comment33 pages, some corrections, bibliography updated

References in corpus (10)

• Gravity coupled with matter and foundation of non-commutative geometry
• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Global quantization of pseudo-differential operators on compact Lie groups, SU(2) and 3-sphere
• CQG algebras: a direct algebraic approach to compact quantum groups
• The Dirac operator on compact quantum groups
• Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory
• Deforming Maps for Lie Group Covariant Creation and Annihilation Operators
• A twisted spectral triple for quantum SU(2)
• Fourier multipliers and group von Neumann algebras
• Hardy-Littlewood inequalities on compact quantum groups of Kac type

Cited by in corpus (3)

• A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces
• $L^p-L^q$ boundedness of Fourier multipliers on quantum Euclidean spaces
• Hörmander type Fourier multiplier theorem and Nikolskii inequality on quantum tori, and applications