# Smooth dense subalgebras and Fourier multipliers on compact quantum groups

## TL;DR

This paper introduces smooth dense subalgebras of compact quantum groups characterized by rapid decay, develops a Fourier analysis framework, and applies it to boundedness and differential calculus extension problems.

## Contribution

It defines new smooth subalgebras using eigenvalues of Dirac-like operators, establishes a Schwartz kernel theorem, and analyzes boundedness and calculus extension on quantum groups.

## Key findings

- Schwartz kernel theorem for operators on compact quantum groups
- Conditions for $L^p-L^q$ boundedness of coinvariant operators
- Necessary and sufficient conditions for calculus extension on quantum SU(2)

## 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.