Given $n\in\mathbb{ N}$ any point on the closed unit disk $\overline{ \mathbb{ D}}$ 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 $\phi:B(\mathcal{ H})\to\mathbb{ C}$, $\{ \phi(U): U\,\mathrm{ unitary}\}=\overline{ \mathbb{ D}}$. We also show that if $\dim$ $\mathcal{ H}$ is finite, for any $w\in\overline{ \mathbb{ D}}$ 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 $\{ \phi(P) : P^*=P^2=P\}=[0,1]$.
Categories: math.FA