A note on \(M\)-ideals in certain algebras of operators (Q1578365)

\textit{N. J. Kalton} and \textit{D. Werner} proved in [Proc. Roy. Soc. Edinburgh Sect. A 125, 493-500 (1995; Zbl 0835.46016)] that, if \(1< p, q<\infty\), the complex Banach space \(X=(\sum_n\ell^n_q)_p\) satisfies that the space \(K(X)\) of compact operators is the only nontrivial \(M\)-ideal in \(L(X)\). In this case both \(X\) and \(X^*\) are uniformly convex, hence the \(M\)-ideals in \(L(X)\) are two-sided closed ideals in \(L(X)\). The authors of the present article investigate the space \(X= (\sum_n\ell^n_1)_p\), \(1< p< \infty\). In this case neither \(X\) nor \(X^*\) is uniformly convex. However, they observe that the proof of Kalton and Werner shows that \(K(X)\) is the only nontrivial \(M\)-ideal in \(L(X)\) which is also a closed ideal in \(L(X)\).