Isomorphisms and projections for quotient-spaces of $\mathscr L_1$-spaces by their reflexive subspaces

S. V. Kislyakov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part VIII, Tome 73 (1977), pp. 91-101

Résumé

Let $Z_1=X_1/E_1$ and $Z_2=X_2/E_2$, where $X_1$ and $X_2$ are $\mathscr L_1$-spaces $E_1\subset X_1$, $E_2\subset X_2$. In this paper we study the following questions: 1) under what conditions are $Z_1$ and $Z_2$ isomorphic; 2) under what conditions is $Z_1$ isomorphic to a complemented subspace of $Z_2$. Some results: (a) if $E_1$ and $E_2$ are reflexive and $Z_1$. is isomorphic to $Z_2$, then one of the spaces E1 E2 is isomorphic to the product of the other by a finite-dimensional space; (b) if $X_1=C(\mathbf T)^*$ ($\mathbf T$ is a circle), $E_1=H^1$ and $E_2$ is reflexive and $X_2=Y^*$ for some $Y$, then it is impossible to imbed $Z_1$ in $Z_2$ as a complemented subspace.