Fourier coefficients of continuous functions on compact groups

Abstract:

Let $G$ be an infinite compact group with dual object $\Sigma$. Letting ${\mathcal {K}_\sigma }$ be the representation space for $\sigma \in \Sigma$, ${\mathcal {E}^2}(\Sigma )$ is the set $\{ A = ({A^\sigma }) \in \Pi \mathcal {B}({\mathcal {K}_\sigma }):\left \| A \right \|_2^2 = {\sum _\sigma }{d_\sigma }\operatorname {Tr} ({A^\sigma }{A^{\sigma *}}) < \infty \}$. For $A \in {\mathcal {E}^2}(\Sigma )$, we show that there is a function $f$ in $C(G)$ such that ${\left \| f \right \|_\infty } \leqslant C{\left \| A \right \|_2}$ and $\operatorname {Tr} (\hat f(\sigma )\hat f{(\sigma )^*}) \geqslant \operatorname {Tr} ({A^\sigma }{A^{\sigma *}})$ for every $\sigma \in \Sigma$.