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\(M\)-ideals in Banach spaces and Banach algebras. (English) Zbl 0789.46011

Lecture Notes in Mathematics. 1547. Berlin: Springer-Verlag. viii, 387 p. (1993).
In 1972 the paper “Structure in real Banach spaces” by E. M. Alfsen and E. G. Effros appeared [Ann. Math., II. Ser. 96, 98-173 (1972; Zbl 0248.46019)]. There an approach was presented by which a number of results from different branches of Banach space geometry (including e.g. \(C^*\)-algebras and Banach lattices) could be treated in a unified way. Roughly speaking the main idea was to study certain subspaces (\(M\)-ideals) and operators (the elements of the centralizer) in an arbitrary Banach space by which in the case of a \(CK\)-space the Banach space under consideration is completely characterized; thus the theory – which is now called \(M\)-structure theory – measures, in a sense, to what extent a given space behaves like a space of continuous functions.
To be more precise, let \(Y\) be a closed subspace of a Banach space \(X\). \(Y\) is called an \(M\)-ideal if the annihilator \(Y^ 0\) of \(Y\) in \(X'\) admits a complementary subspace \(Z\) such that the norm of \(w+z\) equals the sum of the norms of \(w\) and \(z\) for arbitrary \(w\) in \(Y^ 0\) and \(z\) in \(Z\).
\(M\)-deals \(Y\) have interesting approximation properties: every element of \(X\) not contained in \(Y\) has numerous elements of best approximation in \(Y\). More is known if \(X\) is an \(M\)-ideal in its bidual. If this is the case, \(X\) has to have a number of geometric properties (to mention a few, \(X\) is not complemented in \(X''\) unless \(X\) is reflexive, \(X'\) has RNP, \(X\) is a WCG space, and \(X\) has to have Pelczynski’s property \((u)\)). And finally, even stronger conclusions can be made if the space of compact operators \(K(X)\) on \(X\) is an \(M\)-ideal in the space \(L(X)\) of all bounded operators. Then \(X\) is an \(M\)-ideal in \(X''\) (so that the above properties hold), \(X\) has the compact metric approximation property, and on the dual unit sphere the norm and the weak\(^*\)-topology coincide.
By this sample of results which are due to various authors it should be clear why many mathematicians from Banach space theory and related fields have worked in \(M\)-structure theory. Research mainly has concentrated on the following problems:
– Describe all \(M\)-ideals in a given space.
– Characterize the spaces \(X\) such that \(X\) is an \(M\)-ideal in \(X''\).
– Characterize the \(X\) for which \(K(X)\) is an \(M\)-ideal in \(L(X)\).
– Try to apply \(M\)-structure methods.
The aim of the book under review is to provide a systematic presentation of the theory. In particular, it includes Kalton’s characterization of the \(X\) for which \(K(X)\) is an \(M\)-ideal, a result which marks the endpoint of the efforts of many people and which can be regarded as the solution of the last fundamental open problem of the theory.
The book is written to be accessible to everyone who is acquainted with the basic principles of Banach space theory. The contents is as follows:
In chpater I the fundamental definitions, examples and results are given. In particular the authors provide the Alfsen-Effros characterization of \(M\)-ideals by intersection properties.
Chapter II is devoted to deeper results in connection with \(M\)-ideals. For example one finds the lifting theorem of Ando which gives a unified approach to various theorems deaing with extension operators. Spaces for which \(X\) is an \(M\)-ideal in \(X''\) are systematically studied in chapter III. Examples are given not only among the classical Banach, Orlicz and Lorentz spaces but also by considering suitable spaces of analytic functions (e.g. the Bloch spaces). The dual notion to that of an \(M\)- ideal is that of an \(L\)-summand, and thus chapter IV can be regarded as the counterpart of chapter III. It contains a systematic study of spaces \(X\) which are \(L\)-summands in their biduals. Especially interesting are the applications of the theory to problems in harmonic analysis.
Chapter V provides results on \(M\)-ideals and general \(M\)-structure properties in Banach algebras where particular attention is paid to the case of \(C^*\)-algebras. Finally in chapter VI these results are applied to treat the special case of operator algebras. It is this chapter where one finds Kalton’s characterization of the \(X\) with \(K(X)\) an \(M\)- ideal.
The book is very carefully written, and all important results are given with complete proofs; further material is discussed in the notes-and- remarks sections at the end of each chapter.
It is to be expected that the volume will be the standard reference for results in \(M\)-structure theory itself and its applications. Everybody who works in Banach space theory is invited to have a look at it.

MSC:

46B20 Geometry and structure of normed linear spaces
46B25 Classical Banach spaces in the general theory
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46B28 Spaces of operators; tensor products; approximation properties

Citations:

Zbl 0248.46019
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