My first question is the following:
Q1: Let $X$ be a Banach space. If its dual $X^\ast$ is weak* separable, does $X$ admit an infinite-dimensional and separable quotient $X/M$?
To the best of my knowledge, the dual $X^\ast$ is weak* separable, when $X$ satisfies one of the following:
(i) $X$ is separable;
(ii) $X$ is the dual of a separable Banach space;
(iii) $X$ is Hereditary Indecomposable Banach space. That is, every infinite-dimensional closed subspace of $X$ can not be written as a direct sum of two infinite-dimensional closed subspaces.
And, I see that if $X$ satisfies (i) or (ii), $X$ admit an infinite-dimensional and separable quotient.
Q2: Is it true that $X$ admit an infinite-dimensional and separable quotient, if $X$ satisfies (iii) ?
$X^*$is not HI, then $X$ has a separable quotient. Indeed, then by Gowers$X^*$contains a subspace with an unconditional basis and hence, by James, a copy of $c_0$, $\ell_1$, or an infinite dimensional reflexive space. In the last case, $X$ has a reflexive quotient. In the first case, $X$ contains a complemented subspace isomorphic to $\ell_1$ by Bessaga-Pelczynski. In the middle case, $X$ has a quotient isomorphic to either $c_0$ or $\ell_1$ by combining results of Rosenthal and mine and Hagler and mine. $\endgroup$$X^∗$is not HI" should be "If$X^∗$has no HI subspace". $\endgroup$