Regular unitarily invariant spaces on the complex sphere

Let $K$ be a compact space, $X$ a closed subspace of $C(K)$, and $\mu$ a positive measure on $K$. The triple $(X,K,\mu)$ is said to be regular if for any positive function $\varphi\in C(K)$ and for any $\varepsilon>0$ there exists a function $f\in X$ such that $|f|\le\varphi$ on $K$ and $\mu\{t\in K:|f(t)|\ne\varphi(t)\}<\varepsilon$.
The case when $K$ is the unit sphere in $\mathbb C_n$ and the subspace $X$ is invariant with respect to the unitary group is investigated. Sufficient spectral conditions and a necessary condition for regularity are obtained. Connections with compactness of certain Hankel operators and applications to interpolation problems are presented.