\(M\)-ideals of compact operators
A subspace \(E\) of a Banach space \(X\) is called an \(M\)-ideal if the annihilator \(E^ \perp\) of \(E\) in the dual admits a subspace \(F\) such that \(X^*\) is the \(\ell^ 1\)-direct sum of \(E^ \perp\) and \(F\). The investigation of the problem whether for a space \(X\) the space of compact operators \(K(X)\) is an \(M\)-ideal in the bounded operators \(L(X)\) traces back to a paper of Dixmier's published in 1950. Later, after the fundamental contributions of Alfsen and Effros to \(M\)-structure theory, many mathematicians have treated this problem. It was shown by \textit{P. Harmand} and \textit{Å. Lima} [Trans. Am. Math. Soc. 283, 253-264 (1984; Zbl 0545.46009)] that \(K(X)\) is an \(M\)-ideal only if \(X\) satisfies a certain approximation property, and it was open for many years how to find additional conditions which lead to a characterization. The present paper provides this classification for the case of separable spaces. Besides other equivalences it is shown that \(K(X)\) is an \(M\)- ideal in \(L(X)\) iff the following two properties hold: (1) There is a sequence \((K_ n)\) of compact operators such that the \(K_ n\) (resp. the \(K^*_ n\)) tend pointwise to the identity on \(X\) (resp. on \(X^*\)). (2) \(X\) has the property \((M)\), i.e. \(\limsup\| x+ x_ n\|= \limsup\| y+ x_ n\|\) whenever \(\| x\|= \| y\|\) and \((x_ n)\) tends weakly to zero. This theorem has a number of important consequences (concerning e.g. renormings) which are treated in detail. The interested reader is also referred to the monograph of \textit{P. Harmand}, \textit{D. Werner} and \textit{W. Werner} [`\(M\)-ideals in Banach spaces and Banach algebras', Lecture Notes Math. 1547 (1993; Zbl 0789.46011)], where Kalton's result is presented also in the nonseparable setting and where many facts about \(M\)-ideals of compact operators are discussed and to the paper of \textit{ÅLima} in Stud. Math. 113, No. 3, 249-263 (1995) who studies an interesting variant of the property \((M)\).
scientific article; zbMATH DE number 561196