Banach spaces which are $M$-ideals in their biduals

Abstract:

We investigate Banach spaces $X$ such that $X$ is an $M$-ideal in ${X^{{\ast }{\ast }}}$. Subspaces, quotients and ${c_0}$-sums of spaces which are $M$-ideals in their biduals are again of this type. A nonreflexive space $X$ which is an $M$-ideal in ${X^{{\ast }{\ast }}}$ contains a copy of ${c_0}$. Recently Lima has shown that if $K(X)$ is an $M$-ideal in $L(X)$ then $X$ is an $M$-ideal in ${X^{{\ast }{\ast }}}$. Here we show that if $X$ is reflexive and $K(X)$ is an $M$-ideal in $L(X)$, then $K{(X)^{{\ast }{\ast }}}$ is isometric to $L(X)$, i.e. $K(X)$ is an $M$-ideal in its bidual. Moreover, for real such spaces, we show that $K(X)$ contains a proper $M$-ideal if and only if $X$ or ${X^{\ast }}$ contains a proper $M$-ideal.