Some non-commutative averaging theorems

Abstract: 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 $φ:B(\mathcal{H})\to\mathbb{C}$, ${φ(U): U\,\mathrm{ unitary}}=\overline{\mathbb{D}}$. 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 $φ(U)=w$. Lastly, we prove the divisibility property for any state $φ$ on $B(\mathcal{H})$ where $\mathcal{H}$ is infinite-dimensional, showing that ${φ(P) : P^*=P^2=P}=[0,1]$.