Three commuting, unital, completely positive maps that have no minimal dilation

TL;DR

This paper demonstrates the existence of specific examples of three commuting, unital, completely positive maps that lack minimal dilations on any von Neumann algebra, highlighting limitations in dilation theory.

Contribution

It provides the first known examples of such maps with no minimal dilation on any von Neumann algebra, advancing understanding in dilation theory.

Findings

Existence of three commuting, unital, completely positive maps with no minimal dilation

Examples are on type I factors and beyond

Highlights limitations in dilation theory for these maps

Abstract

In this note we prove that there exist at least two examples of three commuting, unital, completely positive maps that have no dilation on a type I factor, and no minimal dilation on any von Neumann algebra.