Finite dimensionality of irreducible unitary representations of compact quantum groups

Abstract

In this paper, we study the representations of Hopf C ∗ {C^ \ast } -algebras; the main result is that every irreducible left unitary representation of a Hopf C ∗ {C^\ast } -algebra with a Haar measure is finite dimensional. To prove this result, we first study the comodule structure of the space of Hilbert-Schmidt operators; then we use this comodule structure to show that every irreducible left unitary representation of a Hopf C ∗ {C^\ast } -algebra with a Haar measure is finite dimensional.