Uniqueness of the maximal ideal of the Banach algebra of bounded operators on \(C([0, \omega_1])\)
continuous functions on the first uncountable ordinal
Denote by \(C([0, \omega_{1}])\) the Banach space of continuous functions on the ordinal interval \([0, \omega_{1}]\), where \(\omega_{1}\) is the first uncountable ordinal. Denote also by \({\mathcal B}(C([0, \omega_{1}]))\) the Banach algebra of bounded operators on \(C([0, \omega_{1}])\). The starting point of the paper under review is a result of \textit{W. Rudin} [Proc. Am. Math. Soc. 8, 39--42 (1957; Zbl 0077.31103)] which implies that each \(T\) in \({\mathcal B}(C([0, \omega_{1}]))\) has a natural representation as a \([0, \omega_{1}] \times [0, \omega_{1}]\)-matrix. Then, it is recalled that \textit{R. J. Loy} and \textit{G. A. Willis} [J. Lond. Math. Soc., II. Ser. 40, No. 2, 327--346 (1989; Zbl 0651.47035)] observed that the set of operators whose final column is continuous when viewed as scalar-valued functions on \([0, \omega_{1}]\) defines a maximal ideal of codimension one in \({\mathcal B}(C([0, \omega_{1}]))\).NEWLINENEWLINEThus, the authors of the present paper give a coordinate-free characterization of this ideal and deduce from it that \({\mathcal B}(C([0, \omega_{1}]))\) contains no other maximal ideals. Then, they obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range. They also investigate the structure of the lattice of all closed ideals of \({\mathcal B}(C([0, \omega_{1}]))\).
scientific article; zbMATH DE number 6044517
scientific article