On ${\ell }_1$-preduals distant by 1

For every predual $X$ of ${\ell }_1$ such that the standard basis in ${\ell }_1$ is weak${}^{\ast }$ convergent, we give explicit models of all Banach spaces $Y$ for which the Banach-Mazur distance $d(X,Y)=1$. As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space ${\ell }_1$, with a predual $X$ as above, has the stable weak${}^{\ast }$ fixed point property if and only if it has almost stable weak${}^{\ast }$ fixed point property, i.e. the dual $Y^{\ast }$ of every Banach space $Y$ has the weak${}^{\ast }$ fixed point property (briefly, $\sigma (Y^{\ast },Y)$-FPP) whenever $d(X,Y)=1$. Then, we construct a predual $X$ of ${\ell }_1$ for which ${\ell }_1$ lacks the stable $\sigma ({\ell }_1,X)$-FPP but it has almost stable $\sigma ({\ell }_1,X)$-FPP, which in turn is a strictly stronger property than the $\sigma ({\ell }_1,X)$-FPP. Finally, in the general setting of preduals of ${\ell }_1$, we give a sufficient condition for almost stable weak${}^{\ast }$ fixed point property in ${\ell }_1$ and we prove that for a wide class of spaces this condition is also necessary.