Uniqueness of the maximal ideal of the Banach algebra of bounded operators on

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• 作者：Tomasz Kania ^{t.kania@lancaster.ac.uk} ; Niels Jakob Laustsen ; ^{n.laustsen@lancaster.ac.uk}
• 关键词：Continuous functions on the first uncountable ordinal ; Bounded operators on Banach spaces ; Maximal ideal ; Loy&ndash ; Willis ideal
• 刊名：Journal of Functional Analysis
• 出版年：2012
• 出版时间：1 June, 2012
• 年：2012
• 卷：262
• 期：11
• 页码：4831-4850
• 全文大小：220 K

文摘

Let be the first uncountable ordinal. A result of Rudin implies that bounded operators on the Banach space of continuous functions on the ordinal interval have a natural representation as -matrices. Loy and Willis observed that the set of operators whose final column is continuous when viewed as a scalar-valued function on defines a maximal ideal of codimension one in the Banach algebra of bounded operators on . We give a coordinate-free characterization of this ideal and deduce from it that 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 .