发表机构
Theoretical Statistics and Mathematics Unit, ISI Banglore（理论统计与数学单元，ISI班加罗尔）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对C*-代数A的稠密子代数A₀的纯无限性转移问题，在A具实秩零、A₀为hfc-闭的假设下解决了该问题，还证明了相关阻碍，并以Cuntz代数的光滑子代数为例验证假设成立。
AI 中文摘要
Pino、Goodearl、Perera和Molina在文献[arxiv.org/abs/0806.4156]中提出的问题8.4：若C*-代数A的稠密子代数A₀作为环是纯无限的，是否能推出A作为C*-代数也是纯无限的？我们将此问题称为纯无限性转移问题，即从环向其C*-完备化转移纯无限性。该问题仍未解决，即使A₀是单位元且单的情况也不例外。我们在两个假设下解决了该问题：A具有实秩零，且A₀在全纯函数演算下是闭的（hfc-闭）。在这些假设下，A₀作为环是纯无限单的当且仅当A作为C*-代数是纯无限单的。我们还证明了一个阻碍：具有非零有限投影的单位元C*-代数，不存在稠密的hfc-闭纯无限单单位子环。最后，这些假设对真子代数成立：对于Cuntz代数𝒪ₙ上的圆周群𝕋的规范作用，光滑子代数𝒪ₙ^∞是真稠密的hfc-闭子代数，且作为环是纯无限单的。
英文摘要
Problem 8.4 of Pino, Goodearl, Perera and Molina in [arxiv.org/abs/0806.4156] asks whether a dense subalgebra $A_{0}$ of a $C^*$-algebra $A$ that is purely infinite as a ring forces $A$ to be purely infinite as a $C^*$-algebra. We call this the $\textit{pure infiniteness transfer problem}$, the transfer being from the ring to its $C^*$-completion. The problem is open, even when $A_{0}$ is unital and simple. We settle it under two hypotheses: $A$ has real rank zero, and $A_{0}$ is closed under holomorphic functional calculus. Under these hypotheses $A_{0}$ is purely infinite simple as a ring if and only if $A$ is purely infinite simple as a $C^*$-algebra. We also prove an obstruction: a unital $C^*$-algebra with a nonzero finite projection has no dense hfc-closed purely infinite simple unital subring. Finally, the hypotheses hold for proper subalgebras, for the gauge action of $\mathbb{T}$ on a Cuntz algebra $\mathcal{O}_n$ , the smooth subalgebra $\mathcal{O}_n^\infty$ is a proper dense hfc-closed subalgebra that is purely infinite simple as a ring.
Comments19 Pages
