Ubiquity of counterexamples to the Smith-Ward problem

Abstract

The Smith-Ward problem about matrix ranges, posed in the 1980s, was recently resolved in the negative by Marcel Scherer (arXiv:2607.04274) by obtaining a three-dimensional operator system $\mathcal{S} \subseteq M_4(C_r^*(\mathbb{F}_2))$ without the lifting property. However, this operator system must be exact. In this paper we show that, for every finitely generated $C^*$-algebra $\mathcal{A}$ without the local lifting property (LLP), there exists a three-dimensional operator system $\mathcal{S} \subseteq M_{n+2}(\mathcal{A})$ without the lifting property (LP), thus generalizing Scherer's result and eliminating the reliance on $\text{Ext}(\mathcal{A})$ not being a group. In particular, we prove that whenever $\mathcal{T}$ is a finite-dimensional operator system without the LP, then $M_{n+2}(C_u^*(\mathcal{T}))$ contains a $3$-dimensional operator system without the LP for some $n \leq 2(\dim(\mathcal{T})-1)$. In this way, we yield a plethora of counterexamples to the Smith-Ward problem. In particular, unlike Scherer's example, these three-dimensional operator systems fail both the LP and exactness. We also prove the existence of a three-dimensional operator system that detects nuclearity for unital $C^*$-algebras, strengthening previous work of Kavruk (J. Funct. Anal., volume 269, 2015).