The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.(See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem).

Definition: A dual Banach space $X^*$is said to be $w^*$-HI if dist ($S_Y$,$S_Z$ )= 0 for every $Y$, $Z$ infinite dimensional $w^*$ closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$ indecomosable if it is not the direct sum of two infinite dimensional w* closed subspaces.

Question I: Does there exist non separable $X$ such that $X^*$ is w*-HI?

Question II: For $X$ HI is the second dual $X^{**}$ w*- inecomposable?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}$= $X$ $\oplus$ $l^2(2^N)$

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.  (See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem.)

Definition: A dual Banach space $X^*$ is said to be $w^*$-HI if $\operatorname{dist} (S_Y,S_Z )= 0$ for every $Y$, $Z$ infinite dimensional $w^*$-closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$-indecomposable if it is not the direct sum of two infinite dimensional $w^*$-closed subspaces.

Question I: Does there exist non separable $X$ such that $X^*$ is $w^*$-HI?

Question II: For $X$ HI is the second dual $X^{**}$ $w^*$-inecomposable?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}= X \oplus l^2(2^N)$.

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.(See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem).

Definition: A dual Banach space $X^*$is said to be $w^*$-HI if dist ($S_Y$,$S_Z$ )= 0 for every $Y$, $Z$ infinite dimensional $w^*$ closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$ indecomosable if its not the direct sum of two infinite dimensional w* closed subspaces.

Question I: Does there exist non separable $X$ such that $X^*$ is w*-HI?

Question II: For $X$ HI is the second dual $X^{**}$ w*- inecomposable?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}$= $X$ $\oplus$ $l^2(2^N)$

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.(See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem).

Definition: A dual Banach space $X^*$is said to be $w^*$-HI if dist ($S_Y$,$S_Z$ )= 0 for every $Y$, $Z$ infinite dimensional $w^*$ closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$ indecomosable if it is not the direct sum of two infinite dimensional w* closed subspaces.

Question I: Does there exist non separable $X$ such that $X^*$ is w*-HI?

Question II: For $X$ HI is the second dual $X^{**}$ w*- inecomposable?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}$= $X$ $\oplus$ $l^2(2^N)$

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.(See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem).

Definition: A dual Banach space $X^*$is said to be w*-HI if dist ($S_Y$,$S_Z$ )= 0 for every $Y$, $Z$ infinite dimensional $w^*$ closed subspaces of $X^*$.

Question I: Does there exist non separable $X$ such that $X^*$ is w*-HI?

Question II: For $X$ HI is the second dual $X^{**}$ w*-HI?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}$= $X$ $\oplus$ $l^2(2^N)$

The following question is inspired by Bill's nice unpublished result that the dual of a non separable Banach space is decomposable.(See the previous posts Decomposable Banach Spaces, Hereditarily indecomposable Banach spaces and Separable Quotient problem and A question related to the separable quotient problem).

Definition: A dual Banach space $X^*$is said to be $w^*$-HI if dist ($S_Y$,$S_Z$ )= 0 for every $Y$, $Z$ infinite dimensional $w^*$ closed subspaces of $X^*$. The space $X^*$ is said to be $w^*$ indecomosable if its not the direct sum of two infinite dimensional w* closed subspaces.

Question I: Does there exist non separable $X$ such that $X^*$ is w*-HI?

Question II: For $X$ HI is the second dual $X^{**}$ w*- inecomposable?

There exists $X$ HI such that $X^*$ is non separable and $X^{**}$= $X$ $\oplus$ $l^2(2^N)$