UHF代数中的交换子
Single commutators in $C^*$-algebras
- College of Science, China University of Petroleum (East China)（中国石油大学（华东）理学院）
- School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文证明在UHF代数中，迹为零的元素均可表示为单个交换子，即核等于所有交换子集合。
中文摘要 AI 辅助
本文证明了UHF代数中每个迹为零的元素都是单个交换子。更精确地，若$(\mathcal{A},\tau)$是带有正规化迹$\tau$的UHF代数，则$\ker\tau=\{[A,B]\colon A,B\in\mathcal{A}\}$。
英文摘要
Let $\mathcal{A}$ be a simple unital $C^*$-algebra with real rank zero, nonempty tracial state space $T(\mathcal{A})$, and strict comparison of projections. We prove that every element $T$ in $\mathcal{A}$ with $τ(T)=0$ for all $τ\in T(\mathcal{A})$ is a single commutator if one of the following conditions holds: $(1)$ $\mathcal{A}$ has unique trace; $(2)$ $\mathcal{A}$ is separable and has stable rank one. We also prove that every nonscalar element in a simple unital purely infinite $C^*$-algebra is a single commutator. Applications include UHF algebras, irrational rotation algebras, and Cuntz algebras.
