The Dixmier property and tracial states for C?-algebras
The Dixmier property and tracial states for C?-algebras
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C?-代数的 Dixmier 性质和迹态
DOI：
10.1016/j.jfa.2017.06.026
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发表时间：
2017
影响因子：
1.7
通讯作者：
Archbold R
中科院分区：
文献类型：
--
作者：
Archbold R
证明了有单位元的C-代数A具有Dixandom性质当且仅当它是弱中心的且满足一定的迹条件。这推广了单C-代数的Haagerup-Zarzó定理。我们还研究了Dixonial性质的一个统一形式，例如vonNeumann代数和幂群的约化C-代数满足Dixonial性质，但不是所有具有Dixonial性质的C-代数都满足Dixonial性质，并且我们得到了具有唯一迹态的单有单位元C-代数具有这种统一性质的充要条件.进一步给出了具有一致Dixlane性质的C-代数的例子，即所有具有Dixlane性质且迹的比较半径有限的C-代数.最后，我们利用一个包含迹数据和代数数值域的公式，确定了任意有单位元的C-代数中两个Dixtons集之间的距离。
It is shown that a unital C⁎-algebra A has the Dixmier property if and only if it is weakly central and satisfies certain tracial conditions. This generalises the Haagerup–Zsidó theorem for simple C⁎-algebras. We also study a uniform version of the Dixmier property, as satisfied for example by von Neumann algebras and the reduced C⁎-algebras of Powers groups, but not by all C⁎-algebras with the Dixmier property, and we obtain necessary and sufficient conditions for a simple unital C⁎-algebra with unique tracial state to have this uniform property. We give further examples of C⁎-algebras with the uniform Dixmier property, namely all C⁎-algebras with the Dixmier property and finite radius of comparison-by-traces. Finally, we determine the distance between two Dixmier sets, in an arbitrary unital C⁎-algebra, by a formula involving tracial data and algebraic numerical ranges.