Idempotents in nest algebras
Idempotents in nest algebras
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DOI：
10.1016/0022-1236(91)90019-2
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发表时间：
1991-04
影响因子：
1.7
通讯作者：
D. Larson;D. Pitts
中科院分区：
文献类型：
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作者：
D. Larson;D. Pitts
完整地刻画了巢代数Alg β中幂等元的代数等价类和相似类。得到了两个幂等等价或相似的充分必要条件。我们的判据给出了病态的例子，并表明对于幂等的每一个等价类，都对应着一个从β × β intoN∪{∞}的“维函数”。通过证明任意维函数对应于一个幂等函数的等价类，完成了代数等价类的刻画。同样，对于每一个维函数序列，都对应一个幂等的交换序列。当一个幂等函数与另一个次幂函数相似时，得到了一个准则。将等价类（或幂等）发送到其相关维函数的映射在巢代数理论中所起的作用类似于将IW * -代数中的投影发送到其中心值迹的映射所起的作用。证明了几乎可交换、相似等幂是同伦的；这与某些c * -代数的情况相反。利用这一点，我们证明了相似的，同时可对角化的幂等函数是同伦的，并且在连续巢情况下，每个对角幂等函数都同伦于一个核心投影。
The algebraic equivalence and similarity classes of idempotents within a nest algebra Alg β are completely characterized. We obtain necessary and sufficient conditions for two idempotents to be equivalent or similar. Our criterion yields examples illustrating pathology and also shows that to each equivalence class of idempotents there corresponds a “dimension function” from β × β intoN∪ { ∞ }. We complete the characterization of the algebraic equivalence classes by proving that any dimension function corresponds to an equivalence class of idempotents. Also, to each sequence of dimension functions, there corresponds a commuting sequence of idempotents. A criterion is obtained for when an idempotent is similar to a subidempotent of another. The mapping which sends an equivalence class (or idempotent) to its associated dimension function plays a role in the nest algebra theory analogous to the role played by the mapping sending a projection in a Type IW∗-algebra to its center valued trace. We prove that almost commuting, similar idempotents are homotopic; this contrasts with the situation in certainC∗-algebras. Using this, we show that similar, simultaneously diagonalizable idempotents are homotopic, and in the continuous nest case, every diagonal idempotent is homotopic to a core projection.