史密斯 - 沃德问题反例的普遍性
Ubiquity of counterexamples to the Smith-Ward problem
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中文总结 AI 辅助
研究史密斯 - 沃德问题，通过对有限生成无局部提升性质的$C^*$ - 代数构造无提升性质的三维算子系统，推广了谢勒结果，产生大量反例，还证明存在检测幺正$C^*$ - 代数核性的三维算子系统。
中文摘要 AI 辅助
20世纪80年代提出的关于矩阵值域的史密斯 - 沃德问题，最近被马塞尔·谢勒否定解决（arXiv:2607.04274），他得到一个无提升性质的三维算子系统$\mathcal{S} \subseteq M_4(C_r^*(\mathbb{F}_2))$，但该算子系统必须是精确的。本文表明，对于每个无局部提升性质（LLP）的有限生成$C^*$ - 代数$\mathcal{A}$，存在一个无提升性质（LP）的三维算子系统$\mathcal{S} \subseteq M_{n + 2}(\mathcal{A})$，推广了谢勒的结果并消除了对$\text{Ext}(\mathcal{A})$不是群的依赖。特别地，证明了只要$\mathcal{T}$是无LP的有限维算子系统，对于某些$n \leq 2(\dim(\mathcal{T}) - 1)$，$M_{n + 2}(C_u^*(\mathcal{T}))$包含一个无LP的三维算子系统。这样就产生了大量史密斯 - 沃德问题的反例，且这些三维算子系统既不满足LP也不精确。还证明了存在一个检测幺正$C^*$ - 代数核性的三维算子系统，加强了卡夫鲁克之前的工作。
英文摘要
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).
