Decomposable subspaces of Banach spaces (original) (raw)

On Hereditarily Indecomposable Banach Spaces

Acta Mathematica Sinica, English Series, 2006

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.

Totally Incomparable Banach Spaces and Three-Space Banach Space Ideals

Mathematische Nachrichten, 1987

In this paper we present four methods to genemte three-space BANACH space ideals. They are based on the concept of total incomparability of H. P. ROSENTH-LL and on a dual concept, total coincomparability, which is here introduced. We use the assertion that the sum of two totally incomptwable closed subspaces of s BANACH space is norm-closed, which is shown by means of an easier and more natural proof than that of ROSEXTHAL [lo], end an analogous property about the total coincompiwability. Several well-known ideals are obtained with the above methods, and so they are three-space ideals.

Linear relations on hereditarily indecomposable normed spaces

Bulletin of the Australian Mathematical Society, 2006

We introduce the notion of hereditarily indecomposable normed space and we prove that this class of normed spaces may be characterised by means of F + and strictly singular linear relations. We also show that if X is a complex hereditarily indecomposable normed space then every partially continuous linear relation in X with dense domain can be written as XI + S, where A 6 C and 5 is a strictly singular linear relation.

M-Ideals in Banach Spaces and Banach Algebras

The present notes centre around the notion of an M-ideal in a Banach space, introduced by E. M. Alfsen and E. G. Effros in their fundamental article "Structure in real Banach spaces" from 1972. The key idea of their paper was to study a Banach space by means of a collection of distinguished subspaces, namely its M-ideals. (For the definition of an M-ideal see Definition 1.1 of Chapter I.) Their approach was designed to encompass structure theories for C *-algebras, ordered Banach spaces, L 1-preduals and spaces of affine functions on compact convex sets involving ideals of various sorts. But Alfsen and Effros defined the concepts of their M-structure theory solely in terms of the norm of the Banach space, deliberately neglecting any algebraic or order theoretic structure. Of course, they thus provided both a unified treatment of previous ideal theories by means of purely geometric notions and a wider range of applicability. Around the same time, the idea of an M-ideal appeared in T. Ando's work, although in a different context. The existence of an M-ideal Y in a Banach space X indicates that the norm of X vaguely resembles a maximum norm (hence the letter M). The fact that Y is an M-ideal in X has a strong impact on both Y and X since there are a number of important properties shared by M-ideals, but not by arbitrary subspaces. This makes M-ideals an important tool in Banach space theory and allied disciplines such as approximation theory. In recent years this impact has been investigated quite closely, and in this book we have aimed at presenting those results of M-structure theory which are of interest in the general theory of Banach spaces, along with numerous examples of M-ideals for which they apply. Our material is organised into six chapters as follows. Chapter I contains the basic definitions, examples and results. In particular we prove the fundamental theorem of Alfsen and Effros which characterises M-ideals by an intersection property of balls. In Chapter II we deal with some of the stunning properties of M-ideals, for example their proximinality. We also show that under mild restrictions M-ideals have to be complemented subspaces, a theorem due to Ando, Choi and Effros. The last section of Chapter II is devoted to an application of M-ideal methods to the classification of L 1-preduals. In Chapter III we investigate Banach spaces X which are M-ideals in their biduals. This geometric assumption has a number of consequences for the isomorphic structure of X. For instance, a Banach space has Pe lczyński's properties (u) and (V) once it is an Mv vi Preface ideal in its bidual; in particular there is the following dichotomy for those spaces X: a subspace of a quotient of X is either reflexive or else contains a complemented copy of c 0. Chapter IV sets out to study the dual situation of Banach spaces which are L-summands in their biduals. The results of this chapter have some possibly unexpected applications in harmonic analysis which we present in Section IV.4. Banach algebras are the subject matter of Chapter V. Here the connections between the notions of an M-ideal and an algebraic ideal are discussed in detail. The most far-reaching results can be proved for what we call "inner" M-ideals of unital Banach algebras. These can be characterised by having a certain kind of approximation of the identity. Luckily the M-ideals which are not inner seem to be the exception rather than the rule. The final Chapter VI presents descriptions of the M-ideals in various spaces of bounded linear operators. In particular we address the problem of which Banach spaces X have the property that the space of compact operators on X is an M-ideal in the space of bounded linear operators, a problem which has aroused a lot of interest since the appearance of the Alfsen-Effros paper. We give two characterisations of those spaces X, one of them following from our work in Chapter V, the other being due to N. Kalton. Each chapter is accompanied by a "Notes and Remarks" section where we try to give precise references and due credits for the results presented in the main body of the text. There we also discuss additional material which is related to the topics of the chapter in question, but could not be included with complete proofs because of lack of space. Only a few prerequisites are indepensable for reading this book. Needless to say, the cornerstones of linear functional analysis such as the Hahn-Banach, Krein-Milman, Krein-Smulian and open mapping theorems are used throughout these notes, often without explicitly mentioning them. We also assume the reader to be familiar with the basics of Banach algebra theory including the Gelfand-Naimark theorem representing a commutative unital C *-algebra in the form C(K), and with various special topics such as the representation of the extreme functionals on a C(K)-space as multiples of Dirac measures or the principle of local reflexivity (an explicit statement of which can be found in Theorem V.1.4). Other concepts that we need but are not so well-known will be recalled as required. For our notation we refer to the list of symbols.

Hereditarily Indecomposable Banach algebras of diagonal operators

Israel Journal of Mathematics, 2011

We provide a characterization of the Banach spaces X with a Schauder basis (en) n∈N which have the property that the dual space X * is naturally isomorphic to the space L diag (X) of diagonal operators with respect to (en) n∈N. We also construct a Hereditarily Indecomposable Banach space X D with a Schauder basis (en) n∈N such that X * D is isometric to L diag (X D) with these Banach algebras being Hereditarily Indecomposable. Finally, we show that every T ∈ L diag (X D) is of the form T = λI + K, where K is a compact operator.

Classification of Ideals in Banach Spaces

Asian Research Journal of Mathematics

Let an operator T belong to an operator ideal J, then for any operators A and B which can be composed with T asBTA then BTA \(\in\) J. Indeed, J contains the class of finite rank Banach Space operators. Now given L(X; Y ). Then J(X; Y ) \(\subseteq\) L(X; Y ) such that J(X; Y ) = {T : X \(\gets\) Y : T \(\subseteq\) }. Thus an operator ideal is a subclass J of L containing every identity operator acting on a one-dimensional Banach space such that: S + T \(\in\) J(X; Y ) where S; T \(\in\) J(X; Y ). If W;Z;X; Y \(\in\) K;A \(\in\) L(W;X);B \(\in\) L(Y;Z) then BTA \(\in\) J(W;Z) whenever T \(\in\) J(X; These properties compare very well with the algebraic notion of ideals in Banach Algebras within whose classes lie compact operators, weakly compact operators, finitely strictly regular operators, completely continuous operators, strictly singular operators among others. Thus, the aim of this paper is to characterize the various classes of ideals in Banach spaces. Special attention is g...

Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

Israel Journal of Mathematics, 1996

Let X be a Banach space with an unconditional finite-dimensional Schauder decomposition (E n). We consider the general problem of characterizing conditions under which one can construct an unconditional basis for X by forming an unconditional basis for each E n. For example, we show that if sup dim E n < ∞ and X has Gordon-Lewis local unconditional structure then X has an unconditional basis of this type. We also give an example of a non-Hilbertian space X with the property that whenever Y is a closed subspace of X with a UFDD (E n) such that sup dim E n < ∞ then Y has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.

On Banach spaces of universal disposition

2016

We present: i) an example of a Banach space of universal disposition that is not separably injective; ii) an example of a Banach space of universal disposition with respect to finite dimensional polyhedral spaces with the Separable Complementation Property; iii) a new type of space of universal disposition nonisomorphic to the previous existing ones.

Linear Relations on Hereditarily Indecomposable Normed Spaces – Corrigendum

Bulletin of the Australian Mathematical Society, 2011

Decomposable subspaces of Banach spaces (original) (raw)

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