Continuous functions on compact groups

Abstract:

We show that every scalar valued continuous function $f(g)$ on a compact group $G$ may be written as $f(g) = \langle \Phi (g) \xi , \eta \rangle$ for all $g \in G$, where $\xi , \eta$ are vectors in a separable Hilbert space $\mathcal {H}$, and $\Phi (g)$ is a strongly continuous unitary valued function on $G$ which is a product of unitary representations and antirepresentations of $G$ on $\mathcal {H}$. This product is countable, but always converges uniformly on $G$. Moreover the supremum norm of $f$ is matched by $\Vert \xi \Vert \Vert \eta \Vert$. This may be viewed as a ‘Fourier product representation’ for $f$, and complements a result of Eymard for the Fourier algebra. For ‘Fourier polynomials’ we show that the Hilbert space $\mathcal {}$ may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka’s duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.