OnM-ideals and best approximation (original) (raw)

Operators ideals and approximation properties

2012

We use the notion of A\AA-compact sets, which are determined by a Banach operator ideal A\AA, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the A\AA-approximation property if the identity map is uniformly approximable on A\AA-compact sets by finite rank operators. The Grothendieck's classic approximation property is the K\KK-approximation property for K\KK the ideal of compact operators and the ppp-approximation property is obtained as the mathcalNp\mathcal N^pmathcalNp-approximation property for mathcalNp\mathcal N^pmathcalNp the ideal of right ppp-nuclear operators. We introduce a way to measure the size of A\AA-compact sets and use it to give a norm on KA\K_\AKA, the ideal of A\AA-compact operators. Most of our results concerning the operator Banach ideal KA\K_\AKA are obtained for right-accessible ideals A\AA. For instance, we prove that KA\K_\AKA is a dual ideal, it is regular and we characterize its maximal hull. A strong concept of approximation property, which makes use of the norm defined on KA\K_\AKA, is also addressed. Finally, we obtain a generalization of Schwartz theorem with a revisited epsilon\epsilonepsilon-product.

Operator Ideals and Approximation Properties

2016

We use the notion of A-compact sets, which are determined by a Banach operator ideal A, to show that most classic results of certain approximation properties and several Banach operator ideals can be systematically studied under this framework. We say that a Banach space enjoys the A-approximation property if the identity map is uniformly approximable on A-compact sets by finite rank operators. The Grothendieck's classic approximation property is the K-approximation property for K the ideal of compact operators and the p-approximation property is obtained as the N p-approximation property for N p the ideal of right p-nuclear operators. We introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Most of our results concerning the operator Banach ideal K A are obtained for right-accessible ideals A. For instance, we prove that K A is a dual ideal, it is regular and we characterize its maximal hull. A strong concept of approximation property, which makes use of the norm defined on K A , is also addressed. Finally, we obtain a generalization of Schwartz theorem with a revisited ǫ-product.

The Banach ideal of -compact operators and related approximation properties

Journal of Functional Analysis, 2013

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and , to show that most classic results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on K A . This notion fits the definition of the A-approximation property, recently introduced by Oja , with K A instead of A. We exemplify the power of the Carl-Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

The Banach ideal of A-compact operators and related approximation properties

Journal of Functional Analysis, 2013

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on K A. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with K A instead of A. We exemplify the power of the Carl-Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

Author's personal copy The Banach ideal of A-compact operators and related approximation properties

We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on K A. This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with K A instead of A. We exemplify the power of the Carl–Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

A new class of operator ideals and approximation numbers

New Trends in Mathematical Science

In this study, we introduce the class of generalized Stolz mappings by generalized approximation numbers. Also we prove that the class of ℓ α p −type mappings are included in the class of generalized Stolz mappings by generalized approximation numbers and we define a new quasinorm equivalent with T α φ (p). Further we give a new class of operator ideals by using generalized approximation numbers and symmetric norming function and we show that this class is an operator ideal.

Approximations of -algebras and the ideal property

Journal of Mathematical Analysis and Applications, 2008

We introduce several classes of C * -algebras (using for this local approximations by "nice" C * -algebras), that generalize the AH algebras, among others. We initiate their study, proving mainly results about the ideal property, but also about the ideals generated by their projections, the real rank zero, the weak projection property, minimal tensor products, extensions, quasidiagonal extensions, ideal structure, the largest ideal with the ideal property and short exact sequences. Some of the previous results of the second named author are generalized.

Ideals of Operators on C∗-Algebras and Their

2016

In this paper we consider we study various classical operator ideals (for instance, the ideals of strictly (co)singular, weakly compact, Dunford-Pettis operators) either on C *-algebras, or preduals of von Neumann algebras.

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.