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 $\mathcal{S}\subseteq M_4(C_r^{\ast }({\mathbb{F}}_2))$ without the lifting property. However, this operator system must be exact. In this paper we show that, for every finitely generated $C^{\ast }$-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 $M_{n+2}(C_u^{\ast }(\mathcal{T}))$ contains a $3$-dimensional operator system without the LP for some $n\le 2(\dim (\mathcal{T})-1)$. 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 $C^{\ast }$-algebras, strengthening previous work of Kavruk (J. Funct. Anal., volume 269, 2015).

15 pages. Comments welcome!

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