Norming subspaces of duals of quotient spaces

Question

In Davis, William J., and William B. Johnson. "Basic sequences and norming subspaces in non-quasi-reflexive Banach spaces." Israel Journal of Mathematics 14.4 (1973): 353-367., the authors discuss the following problem

"If $M$ is a norming subspace of $X^*$ and $Y$ is an $M$-closed subspace of $X$, then is $M \cap Y^\perp$ a norming subspace of $(X/Y)^*$ (where $Y^\perp$ is identified with $(X/Y)^*$ in the canonical way)?"

In the paper, the authors conclude that if $X$ is not quasi-reflexive, then there exists a norming subspace $M$ such that the conclusion fails.

What I am interested in is whether the result can be salvaged in some way by adding some reasonable conditions on $M$, and I would like to know if there is anything in the literature that has addressed this. The papers citing this paper do not discuss this particular question any further.

Thank you for your time.

Answer 1

Some comments on the problem (I print them as an answer, because they are too lengthy for a comment):

(1) As I understand, the condition that $Y$ is $M$-closed is equivalent to the condition that $M\cap Y^\perp$ is total over the quotient $X/Y$.

(2) So, if $X/Y$ is quasireflexive, then "$M\cap Y^\perp$ is total" implies "$M\cap Y^\perp$ is norming" (we do not need quasireflexivity of $X$ itself).

(3) You would like to get a condition on $M$ under which for any subspace $Y$ in $X$ the condition "$M\cap Y^\perp$ is total" implies "$M\cap Y^\perp$ is norming".

(4) There is a trivial condition of this type: $M$ is of finite codimension in $X^*$ (but I doubt that it is satisfactory for you).

(5) It is tempting to claim that any $M$ of infinite codimension in $X^*$ fails to satisfy this condition for some $Y$, but this is not true. If $X$ itself is a dual space and $M$ is its predual, then $M$ is of infinite codimension in $X^*$ for non-quasireflexive $X$, on the other hand, the condition from (3) is satisfied. In fact, if $Y$ is $M$-closed, then $X/Y$ is the dual of $Y^\perp\cap M$, and so "$Y^\perp\cap M$ is norming over $X/Y$"

(6) Of course, one can investigate this matter further. My survey Weak* sequential closures in Banach space theory and their applications could be helpful for this. The survey was written in 1999, but I did not notice any further work in this direction except for my note Weak$^*$ closures and derived sets in dual Banach spaces. Possibly I missed something because I do not work in this direction now.