Some non-commutative averaging theorems

Given $n\in \mathbb{N}$ any point on the closed unit disk $\bar{\mathbb{D}}$ can be written as the average of $n$ points on the unit circle ${\mathbb{S}}^1$. Here we discuss a non-commutative version of this result. We prove that for any Hilbert space $\mathcal{H}$ and a state $\phi :B(\mathcal{H})\to \mathbb{C}$, $\{\phi (U):U\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}\}=\bar{\mathbb{D}}$. We also show that if $\dim$ $\mathcal{H}$ is finite, for any $w\in \bar{\mathbb{D}}$ we can choose a unitary $U$ with atmost $3$ distinct eigenvalues such that $\phi (U)=w$. Lastly, we prove the divisibility property for any state $\phi$ on $B(\mathcal{H})$ where $\mathcal{H}$ is infinite-dimensional, showing that $\{\phi (P):P^{\ast }=P^2=P\}=[0,1]$.