We characterize Banach spaces \(X\) satisfying the relation \[ L(X)^*=(X^*{\hat \otimes}_{\pi}X^{**})\oplus_ 1(X{\hat \otimes}_{\epsilon}X^*)^{\perp} \] in terms of the asymptotic behaviour of certain nets of finite dimensional operators converging strongly to the identity on \(X\).
Central for our considerations is the concept of an inner \(M\)-ideal, which to some extend is investigated systematically. The results thus obtained allow also to describe the \(M\)-ideals in some algebras of bounded operators explicitly. For example, it is shown that \(J\) is an \(M\)-ideal of \(L({\mathfrak A})\), the space of bounded operators on a function algebra \({\mathfrak A}\), if and only if \[ J=\{T\in L({\mathfrak A})| \quad \limsup_{k\to D}\| T^*\delta_ k\| =0\}, \] where \(D\) runs through the \(p\)-sets of \({\mathfrak A}\). An important tool in our reasoning is a - seemingly unknown - result stating that any operator on \({\mathfrak A}\) with an ‘almost real’ numerical range is the perturbation of a multiplication operator.