A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

On relations between weak approximation properties and their inheritances to subspaces

It is shown that for the separable dual X^∗ of a Banach space X, if X^∗ has the weak approximation property, then X^∗ has the metric weak approximation property. We introduce the properties W^∗D and MW^∗D for Banach spaces. Suppose that M is a closed subspace of a Banach space X such that M^⊥ is complemented in the dual space X^∗, where for all m∈M}. Then it is shown that if a Banach space X has the weak approximation property and W^∗D (respectively, metric weak approximation property and MW^∗D), then M has the weak approximation property (respectively, bounded weak approximation property).     

On dual and three space problems for the compact approximation property

We introduce the properties W^∗D and BW^∗D for the dual space of a Banach space. And then solve the dual problem for the compact approximation property (CAP): if X^∗ has the CAP and the W^∗D, then X has the CAP. Also, we solve the three space problem for the CAP: for example, if M is a closed subspace of a Banach space such that M^⊥ is complemented in X^∗ and X^∗ has the W^∗D, then X has the CAP whenever X/M has the CAP and M has the bounded CAP. Corresponding problems for the bounded compact approximation property are also addressed.     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.     

Sigma-locally uniformly rotund and sigma-weak Kadets dual norms

The dual X^∗ of a Banach space X admits a dual σ-LUR norm if (and only if) X^∗ admits a σ-weak^∗ Kadets norm if and only if X^∗ admits a dual weak^∗ LUR norm and moreover X is σ-Asplund generated.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Genericity and amalgamation of classes of Banach spaces

We study universality problems in Banach space theory. We show that if A is an analytic class, in the Effros-Borel structure of subspaces of C([0,1]), of non-universal separable Banach spaces, then there exists a non-universal separable Banach space Y, with a Schauder basis, that contains isomorphs of each member of A with the bounded approximation property. The proof is based on the amalgamation technique of a class C of separable Banach spaces, introduced in the paper. We show, among others, that there exists a separable Banach space R not containing L¹(0,1) such that the indices β and rND are unbounded on the set of Baire-1 elements of the ball of the double dual R∗∗ of R. This answers two questions of H.P. Rosenthal.We also introduce the concept of a strongly bounded class of separable Banach spaces. A class C of separable Banach spaces is strongly bounded if for every analytic subset A of C there exists Y∈C that contains all members of A up to isomorphism. We show that several natural classes of separable Banach spaces are strongly bounded, among them the class of non-universal spaces with a Schauder basis, the class of reflexive spaces with a Schauder basis, the class of spaces with a shrinking Schauder basis and the class of spaces with Schauder basis not containing a minimal Banach space X.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

Banach spaces in various positions

We formulate a general theory of positions for subspaces of a Banach space: we define equivalent and isomorphic positions, study the automorphy index a(Y,X) that measures how many non-equivalent positions Y admits in X, and obtain estimates of a(Y,X) for X a classical Banach space such as ?p,Lp,L₁,C(ωω) or C[0,1]. Then, we study different aspects of the automorphic space problem posed by Lindenstrauss and Rosenthal; namely, does there exist a separable automorphic space different from c₀ or ?₂? Recall that a Banach space X is said to be automorphic if every subspace Y admits only one position in X; i.e., a(Y,X)=1 for every subspace Y of X. We study the notion of extensible space and uniformly finitely extensible space (UFO), which are relevant since every automorphic space is extensible and every extensible space is UFO. We obtain a dichotomy theorem: Every UFO must be either an L_∞-space or a weak type 2 near-Hilbert space with the Maurey projection property. We show that a Banach space all of whose subspaces are UFO (called hereditarily UFO spaces) must be asymptotically Hilbertian; while a Banach space for which both X and X^∗ are UFO must be weak Hilbert. We then refine the dichotomy theorem for Banach spaces with some additional structure. In particular, we show that an UFO with unconditional basis must be either c₀ or a superreflexive weak type 2 space; that a hereditarily UFO Köthe function space must be Hilbert; and that a rearrangement invariant space UFO must be either L_∞ or a superreflexive type 2 Banach lattice.     

Exceptional family of elements and solvability of variational inequalities for mappings defined only on closed convex cones in Banach spaces

In this paper, we generalize the concept of exceptional family of elements for a completely continuous field from Hilbert spaces to Banach spaces and we study the solvability of the variational inequalities with respect to a mapping f that is from a closed convex cone of a Banach space B to the dual space B^∗ by applying the generalized projection operator πK and by using the Leray-Schauder type alternative.     

Borel properties of linear operators

Given an injective bounded linear operator T:X→Y between Banach spaces, we study the Borel measurability of the inverse map T−1:TX→X. A remarkable result of Saint-Raymond (Ann. Inst. Fourier (Grenoble) 26 (1976) 211-256) states that if X is separable, then the Borel class of T⁻¹ is α if, and only if, X∗ is the αth iterated sequential weak∗-closure of T∗Y∗ for some countable ordinal α. We show that Saint-Raymond's result holds with minor changes for arbitrary Banach spaces if we assume that T has certain property named co-σ-discreteness after Hansell (Proc. London Math. Soc. 28 (1974) 683-699). As an application, we show that the Borel class of the inverse of a co-σ-discrete operator T can be estimated by the image of the unit ball or the restrictions of T to separable subspaces of X. Our results apply naturally when X is a WCD Banach space since in this case any injective bounded linear operator defined on X is automatically co-σ-discrete.     

Complementably universal Banach spaces, II

The two main results are:

A.

If a Banach space X is complementably universal for all subspaces of c₀ which have the bounded approximation property, then X^∗ is non-separable (and hence X does not embed into c₀).

B.

There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.

Theorem B solves a problem that dates from the 1970s.     

Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

Approximation of the limit distance function in Banach spaces

In this paper we study the behavior of the limit distance function d(x)=limdist(x,Cn) defined by a nested sequence (Cn) of subsets of a real Banach space X. We first present some new criteria for the non-emptiness of the intersection of a nested sequence of sets and of their ε-neighborhoods from which we derive, among other results, Dilworth's characterization [S.J. Dilworth, Intersections of centred sets in normed spaces, Far East J. Math. Sci. (Part II) (1988) 129-136 (special volume)] of Banach spaces not containing c₀ and Marino's result [G. Marino, A remark on intersection of convex sets, J. Math. Anal. Appl. 284 (2003) 775-778]. Passing then to the approximation of the limit distance function, we show three types of results: (i) that the limit distance function defined by a nested sequence of non-empty bounded closed convex sets coincides with the distance function to the intersection of the weak^∗-closures in the bidual; this extends and improves the results in [J.M.F. Castillo, P.L. Papini, Distance types in Banach spaces, Set-Valued Anal. 7 (1999) 101-115]; (ii) that the convexity condition is necessary; and (iii) that in spaces with separable dual, the distance function to a weak^∗-compact convex set is approximable by a (non-necessarily nested) sequence of bounded closed convex sets of the space.     

On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property

In this paper, we prove that a Banach space X and its dual space X^∗ have uniform normal structure if . The García-Falset coefficient R(X) is estimated by the CNJ(X)-constant and the weak orthogonality coefficient introduced by B. Sims. Finally, we present an affirmative answer to a conjecture by L. Maligranda concerning the relation between the James and CNJ(X)-constants for a Banach space.     

Some remarks on proximinality in higher dual spaces

In this paper we consider proximinality questions for higher ordered dual spaces. We show that for a finite dimensional uniformly convex space X, the space C(K,X) is proximinal in all the duals of even order. For any family of uniformly convex Banach spaces {Xα}_{α∈Γ} we show that any finite co-dimensional proximinal subspace of X=c_0⊕Xα is strongly proximinal in all the duals of even order of X.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

James orthogonality and orthogonal decompositions of Banach spaces

We establish decompositions of a uniformly convex and uniformly smooth Banach space B and dual space B^∗ in the form B=M?J^∗M^⊥ and B^∗=M^⊥?JM, where M is an arbitrary subspace in B, M^⊥ is its annihilator (subspace) in B^∗, J:B→B^∗ and J^∗:B^∗→B are normalized duality mappings. The sign ? denotes the James orthogonal summation (in fact, it is the direct sums of the corresponding subspaces and manifolds). In a Hilbert space H, these representations coincide with the classical decomposition in a shape of direct sum of the subspace M and its orthogonal complement M^⊥: H=M⊕M^⊥.