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On the M-structure of spaces of bounded operators and Banach algebras. (English) Zbl 0694.46014

Berlin: Freie Univ., Diss. 103 p. (1988).
This doctoral dissertation contains a number of important results on the M-structure of operator spaces. Following E. M. Alfsen and E. G. Effros [Ann. Math., II. Ser. 96, 98-173 (1972; Zbl 0248.46019)] a subspace J of a Banach space E is called an M-ideal if its annihilator \(J^{\perp}\) admits an \(\ell^ 1\)-direct complement in \(E^*:\) \(E^*=J^{\perp}\oplus_ 1J^{\#}\). It depends on whether or not \(J^{\#}\) is weak* closed if J admits an \(\ell^{\infty}\)-direct complement in E; in this case J will be called an M-summand.
The first part of the author’s thesis contains basic definitions and auxiliary results to be used in the subsequent chapters. In Chapter 2 it is proved that an M-summand in the space L(X,Y) of bounded linear operators from X into Y necessarily has the form L(X,J), J an M-summand in Y, provided \(X^*\) has no M-summands apart from the trivial ones (which are \(X^*\) and \(\{\) \(0\})\). Also, the corresponding description in the case that the M-summands of Y rather than \(X^*\) are trivial is obtained. As a matter of fact, the author’s results are more general than outlined above, they involve several concepts from M-structure theory such as the centralizer of a Banach space. We will not go into the details of the defnition of the centralizer but just mention that it is an algebra of bounded linear operators which behave in some sense like multiplication operators on function spaces.
It is this operator algebra which is the subject-matter of the third chapter. Here a theorem of Bishop-Stone-Weierstraß type is proved; as a corollary the author obtains a characterization of \(C_{\sigma}\)- spaces by a richness condition on the centralizer. Moreover, the centralizer of an injective tensor product is described in terms of the centralizers of the factors. This description makes use of a topological construction called the k-product.
The final chapter contains the most interesting results. Here a special class of M-ideals in Banach algebras, which the author calls “inner M- ideals”, is investigated. (These are M-ideals induced by multiplication operators in the bidual Banach algebra, equipped with one of the Arens products.) It is pointed out that the ideal of compact operators K(X) automatically is an inner M-ideal once it is an M-ideal in L(X). This leads to a characterization of those Banach spaces X for which K(X) is an M-ideal in L(X) in terms of a strengthened version of the metric compact approximation property. Also, it is shown that all the M-ideals in L(X) are inner M-ideals for function algebras X, and a complete description of the M-ideals in \(L(L^ 1(\mu))\) is given.

MSC:

46B20 Geometry and structure of normed linear spaces
47L05 Linear spaces of operators
46A32 Spaces of linear operators; topological tensor products; approximation properties

Citations:

Zbl 0248.46019
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