Some non-commutative averaging theorems
Saptak Bhattacharya
TL;DR
This paper investigates the non-commutative analogue of averaging unit complex numbers by studying the image of unitaries and projections under states on operator algebras. It proves that in finite dimension any $w$ with $|w|\le1$ is realized by a unitary with at most 3 eigenvalues, while in infinite dimensions the set $\{\phi(U):U\in U(\mathcal{H})\}=\overline{\mathbb{D}}$ and $\{\phi(P):P^*=P\}= [0,1]$ (divisibility). It discusses implications for normal vs non-normal states and related norm-based corollaries. The results illuminate how non-commutative probability measures mimic Lebesgue-like divisibility properties and provide constructive spectral realizations of prescribed values.
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 $\{φ(P) : P^*=P^2=P\}=[0,1]$.