Four-dimensional operator systems without the lifting property

TL;DR

This paper constructs explicit 4-dimensional operator systems within the Calkin algebra that lack the lifting property, revealing limitations in lifting unital completely positive maps in certain operator algebra contexts.

Contribution

It provides explicit examples of 4-dimensional operator systems without the lifting property, linked to unital $C^*$-algebras generated by unitaries, advancing understanding of liftability issues.

Findings

Explicit 4-dimensional operator systems without the lifting property in the Calkin algebra

Counterexamples to the generalized Smith-Ward problem for three self-adjoint operators

Demonstration of limitations in lifting unital completely positive maps

Abstract

The purpose of this note is to provide a family of explicit examples of $4$-dimensional operator systems contained in the Calkin algebra $\mathcal{Q}(\mathcal{H})$ on a separable infinite-dimensional Hilbert space $\mathcal{H}$ for which the identity map has no unital completely positive (ucp) lift to $\mathcal{B}(\mathcal{H})$ with respect to the canonical quotient map $\pi:\mathcal{B}(\mathcal{H}) \to \mathcal{Q}(\mathcal{H})$. More specifically, to each unital $C^*$-algebra $\mathcal{A}$ generated by $n$ unitaries and unital $*$-homomorphism $\rho:\mathcal{A} \to \mathcal{Q}(\mathcal{H})$ with no ucp lift, we construct a four-dimensional operator subsystem $\mathcal{S}$ of $M_{n+1}(\mathcal{A})$ without the lifting property. As a result, for each $n \geq 2$ we exhibit a four-dimensional operator system $\mathcal{S}$ in $M_{n+1}(C_r^*(\mathbb{F}_n))$ without the lifting property. We also obtain explicit examples where the generalized Smith-Ward problem for liftings of joint matrix ranges for three self-adjoint operators has a negative answer.