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

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