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Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support


Authors: William B. Johnson, Tomasz Kania and Gideon Schechtman
Journal: Proc. Amer. Math. Soc. 144 (2016), 4471-4485
MSC (2010): Primary 46H10, 47B38, 47L10; Secondary 06F30, 46B26, 47L20
DOI: https://doi.org/10.1090/proc/13084
Published electronically: April 25, 2016
MathSciNet review: 3531195
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Abstract: Let $ \lambda $ be an infinite cardinal number and let $ \ell _\infty ^c(\lambda )$ denote the subspace of $ \ell _\infty (\lambda )$ consisting of all functions that assume at most countably many non-zero values. We classify all infinite-dimensional complemented subspaces of $ \ell _\infty ^c(\lambda )$, proving that they are isomorphic to $ \ell _\infty ^c(\kappa )$ for some cardinal number $ \kappa $. Then we show that the Banach algebra of all bounded linear operators on $ \ell _\infty ^c(\lambda )$ or $ \ell _\infty (\lambda )$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws' approach description of the lattice of all closed ideals of $ \mathscr {B}(X)$, where $ X = c_0(\lambda )$ or $ X=\ell _p(\lambda )$ for some $ p\in [1,\infty )$, and we classify the closed ideals of $ \mathscr {B}(\ell _\infty ^c(\lambda ))$ that contains the ideal of weakly compact operators.


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Additional Information

William B. Johnson
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: johnson@math.tamu.edu

Tomasz Kania
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-956 Warszawa, Poland; Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom —and — School of Mathematical Sciences, Western Gateway Building, University College Cork, Cork, Ireland
Email: tomasz.marcin.kania@gmail.com

Gideon Schechtman
Affiliation: Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel
Email: gideon@weizmann.ac.il

DOI: https://doi.org/10.1090/proc/13084
Received by editor(s): February 11, 2015
Received by editor(s) in revised form: December 10, 2015, and January 5, 2016
Published electronically: April 25, 2016
Additional Notes: The first-named author was supported in part by NSF DMS-1301604 and the U.S.-Israel Binational Science Foundation
The second-named author was supported by the Warsaw Centre of Mathematics and Computer Science
The third-named author was supported in part by the U.S.-Israel Binational Science Foundation
Part of this work was done when the third-named author was participating in the NSF Workshop in Analysis and Probability held at Texas A&M University
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2016 American Mathematical Society