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# Character Density in Central Subalgebras of Compact Quantum Groups

Published:2017-04-24
Printed: Sep 2017
• Mahmood Alaghmandan,
Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg SE-412 96, Sweden
• Jason Crann,
School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6
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## 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 $L^1(\mathbb{G})$.
 Keywords: compact quantum group, irreducible character
 MSC Classifications: 43A20 - $L^1$-algebras on groups, semigroups, etc. 43A40 - Character groups and dual objects 46J40 - Structure, classification of commutative topological algebras

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