Blanco, A. A result on the ideal structure of \(\mathcal L^r(X)\) for uniformly convex \(X\). (English) Zbl 1420.46022 Positivity 23, No. 2, 485-491 (2019). The main result of this paper is that, within the class of uniformly convex Banach lattices, the purely atomic ones are completely characterized by the fact that every positive compact operator on them is approximable regular. Reviewer: Ömer Gök (Istanbul) MSC: 46B28 Spaces of operators; tensor products; approximation properties 46B42 Banach lattices 47B65 Positive linear operators and order-bounded operators 47B07 Linear operators defined by compactness properties 47L10 Algebras of operators on Banach spaces and other topological linear spaces Keywords:Banach lattice; compact operator; order ideal; regular operator; order continuous norm; atomic space PDFBibTeX XMLCite \textit{A. Blanco}, Positivity 23, No. 2, 485--491 (2019; Zbl 1420.46022) Full Text: DOI References: [1] Abramovich, Y.A., Aliprantis, C.D.: Positive Operators. Handbook of the Geometry of Banach spaces, vol. I, pp. 85-122. North-Holland, Amsterdam (2001) · Zbl 1202.47042 [2] Aliprantis, C.D., Burkinshaw, O.: Positive Operators, Pure and Applied Mathematics, vol. 119. Academic Press, Orlando, FL (1985) · Zbl 0608.47039 [3] Chen, Z.L., Wickstead, A.W.: The order properties of \[r\] r-compact operators on Banach lattices. Acta Math. Sin. (Engl. Ser.) 23(3), 457-466 (2007) · Zbl 1123.46015 · doi:10.1007/s10114-005-0783-2 [4] Fremlin, D.H.: A positive compact operator. Manuscr. Math. 15(4), 323-327 (1975) · Zbl 0318.47013 · doi:10.1007/BF01486602 [5] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Function Spaces, Results in Mathematics and Related Areas, vol. 97. Springer, Berlin (1979) · Zbl 0403.46022 [6] Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974) · Zbl 0296.47023 · doi:10.1007/978-3-642-65970-6 [7] Wickstead, A.W.: Positive compact operators on Banach lattices: some loose ends. Positivity and its applications (Ankara, 1998). Positivity 4(3), 313-325 (2000) · Zbl 0984.47032 · doi:10.1023/A:1009838931430 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.