Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C ( [ 0 , ? 1 ] )
Uniqueness of the maximal ideal of the Banach algebra of bounded operators on C ( [ 0 , ? 1 ] )
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C ([ 0 , ? 1 ] ) 上有界算子的 Banach 代数最大理想的唯一性
DOI：
10.1016/j.jfa.2012.03.011
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发表时间：
2012
影响因子：
1.7
通讯作者：
Kania T
中科院分区：
文献类型：
--
作者：
Kania T
设ω 1是第一个不可数序数。Rudin的一个结果暗示了序区间[0，ω1]上连续函数的Banach空间C（[0，ω 1]）上的有界算子有一个自然表示为[0，ω1]×[0，ω1]-矩阵. Loy和Willis观察到，当被看作[0，ω1]上的标量值函数时，其最后一列连续的算子集定义了C（[0，ω 1]）上有界算子的Banach代数B（C（[0，ω 1]））中余维为1的极大理想。我们给出了这个理想的一个与坐标无关的特征，并由此推出B（C（[0，ω1]））不包含其它极大理想.然后给出了描述可分值域算子的严格小理想的等价条件，最后讨论了B（C（[0，ω1]））的闭理想格的结构.
Let ω1be the first uncountable ordinal. A result of Rudin implies that bounded operators on the Banach space C([0,ω1]) of continuous functions on the ordinal interval [0,ω1] have a natural representation as [0,ω1]×[0,ω1]-matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on [0,ω1] defines a maximal ideal of codimension one in the Banach algebra B(C([0,ω1])) of bounded operators on C([0,ω1]). We give a coordinate-free characterization of this ideal and deduce from it that B(C([0,ω1])) contains no other maximal ideals. We then obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range, and finally we investigate the structure of the lattice of all closed ideals of B(C([0,ω1])).