Character density in central subalgebras of compact quantum groups

TL;DR

This paper extends classical Fourier analysis density results to the setting of compact quantum groups, establishing the density of characters in various subalgebras and partially answering an open question by Woronowicz.

Contribution

It proves the density of characters in fixed point spaces and centers of quantum group algebras, and characterizes the center of $L^1(G)$ for many quantum groups, including Kac algebras.

Findings

Characters are dense in fixed point spaces of conjugation actions.

Characters are weak* and norm dense in the centers of certain algebras.

The center of $L^1(G)$ is spanned by quantum characters for many quantum groups.

Abstract

We investigate quantum group generalizations of various density results from Fourier analysis on compact groups. In particular, we establish the density of characters in the space of fixed points of the conjugation action on $L^2(\mathbb{G})$, and use this result to show the weak* density and norm density of characters in $ZL^{\infty}(\mathbb{G})$ and $ZC(\mathbb{G})$, respectively. As a corollary, we partially answer an open question of Woronowicz. At the level of $L^1(\mathbb{G})$, we show that the center $\mathcal{Z}(L^1(\mathbb{G}))$ is precisely the closed linear span of the quantum characters for a large class of compact quantum groups, including arbitrary compact Kac algebras. In the latter setting, we show, in addition, that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented $\mathcal{Z}(L^1(\mathbb{G}))$-submodule of $L^1(\mathbb{G})$.