Some non-commutative averaging theorems

TL;DR

This paper extends classical averaging results to a non-commutative setting involving Hilbert spaces and states on operator algebras, demonstrating how complex numbers and real intervals can be represented through unitary operators and projections.

Contribution

It introduces a non-commutative averaging theorem for unitaries and projections in operator algebras, generalizing classical geometric averaging results.

Findings

The set of values of a state on unitaries covers the closed disk.

Any point in the disk can be achieved by a unitary with at most 3 eigenvalues in finite dimensions.

The set of possible values of a state on projections is the entire interval [0,1].

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 $\phi:B(\mathcal{H})\to\mathbb{C}$, $\{\phi(U): U\,\mathrm{ unitary}\}=\overline{\mathbb{D}}$. 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 $\{\phi(P) : P^*=P^2=P\}=[0,1]$.