THE QUOTIENT ALGEBRA OF COMPACT-BY-APPROXIMABLE OPERATORS ON BANACH SPACES FAILING THE APPROXIMATION PROPERTY

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.

Fixed-point-free embeddings of graphs in their complements

The following is proved: IfGis a labeled(p,p−2)graph wherep≥2, then there exists an isomorphic embeddingϕofGin its complementG¯such thatϕhas no fixed vertices. The extension to(p,p−1)graphs is also considered.