Structure of total subspaces of dual Banach spaces

Abstract

Let $X$ be a separable nonquasireflexive Banach space. Let $Y$ be a Banach space isomorphic to a subspace of $X^*$. The paper is devoted to the following questions: 1. Under what conditions does there exist an isomorphic embedding $T:Y\to X^*$ such that subspace $T(Y)\subset X^*$ is total? 2. If such embeddings exist, what are the possible orders of $T(Y)$? Here we need to recall some definitions. For a subset $M\subset X^*$ we denote the set of all limits of weak$^*$ convergent sequences in $M$ by $M_{(1)}$. Inductively, for ordinal number $\alpha$ we let $$M_{(\alpha)}=\cup_{\beta<\alpha}(M_{(\beta)})_{(1)}.$$ The least ordinal $\alpha$ for which $M_{(\alpha)}= M_{(\alpha+1)}$ is called the {\it order} of $M$.