1 results
THE QUOTIENT ALGEBRA OF COMPACT-BY-APPROXIMABLE OPERATORS ON BANACH SPACES FAILING THE APPROXIMATION PROPERTY
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 110 / Issue 2 / April 2021
- Published online by Cambridge University Press:
- 07 June 2019, pp. 266-288
- Print publication:
- April 2021
-
- Article
- Export citation
-
We initiate a study of structural properties of the quotient algebra
${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces 
$X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from 
$c_{0}$ into 
${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where 
$Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space 
$c_{0}(\unicode[STIX]{x1D6E4})$ into 
${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where 
$Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.
