On complex structures and uniqueness of algebra norms in Banach spaces

For $X$ an infinite dimensional Banach space, we contribute to the study of the Banach algebra $L(X)\mathrm{/}S(X)$, where $S(X)$ is the ideal of strictly singular operators. We extend results of Ferenczi-Galego (2007) by proving that $\mathrm{\parallel }I-J{\mathrm{\parallel }}_S\ge 2$, whenever $I$ is a complex structure on a real space $X$ and $J$ extends a complex structure on a hyperplane of $X$, and where $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_S$ denotes a certain algebra norm on $L(X)\mathrm{/}S(X)$ dominated by the usual quotient norm $\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }$. We solve two questions of Kalton-Swanson (1982) by proving that if $X=Z_2$ the Kalton-Peck space, then $L(Z_2)\mathrm{/}S(Z_2)$ a) is not complete for $\mathrm{\parallel }\mathrm{.}{\mathrm{\parallel }}_S$ and b) that it is not *-isomorphic to a $C^{\ast }$-algebra for $\mathrm{\parallel }\mathrm{.}\mathrm{\parallel }$. In particular $L(Z_2)\mathrm{/}S(Z_2)$ admits two inequivalent *-algebra norms.