This is an announcement for the paper "Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support" by William B. Johnson, Tomasz Kania, and Gideon Schechtman.
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 which 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 contain the ideal of weakly compact operators.
Archive classification: math.FA
Submitted from: tomasz.marcin.kania@gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1502.03026
or