Åsvald Lima

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Peŀczyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.

THE METRIC APPROXIMATION PROPERTY, NORM-ONE PROJECTIONS AND INTERSECTION PROPERTIES OF BALLS

We show that ifE is a Banach space with the Radon-Nikodym property thenE has the metric approximation property if and only if the space of finite rank operators is locally complemented in the space of bounded operators.

Ideals of compact operators

We give an example of a Banach space such that is not an ideal in . We prove that if is a weak denting point in the unit ball of and if is a closed subspace of a Banach space , then the set of norm-preserving extensions of a functional is equal to the set . Using this result, we show that if is an -ideal in and is a reflexive Banach space, then is an -ideal in whenever is an ideal in . We also show that is an ideal (respectively, an -ideal) in for all Banach spaces whenever is an ideal (respectively, an -ideal) in and has the compact approximation property with conjugate operators.

The structure of finite dimensional Banach spaces with the 3.2. Intersection property

Let X be a Banach space over the real numbers. Let n and k be integers with 2 ~< k < n. We say tha t X has the n.k. intersection property (n.k.I.P.) if the following holds: Any n balls in X intersect provided any k of them intersect. In [2], O. I-Ianner characterized finite dimensional spaces with the 3.2.I.P. by the facial structure of their unit hall. He also proved tha t this property is preserved under 11and /oo-summands, i.e. direct sums X | Y with the y n o r m ]]xli + HY]I or the lop-norm max (llxl], ]] y]]). We shall prove the converse of this result. Any finite dimensional Banach space X with the 3.2.I.P. is obtained from the real line by repeated 11and/~-summands . t tanner proved this for dimension a t most 5. In sections 2 to 4 we gradually introduce the concepts and theorems tha t we need. To become familiar with the techniques involved, we have included the proof of some of the results. In sections 5 and 6 we prove some technical lemmas and characterize the parallel-faces and split-faces among the faces of the unit balls of Banach spaces with the 3.2.I.P. These results are used in the proof of the main result in section 7. Banach spaces are denoted X, Y, and Z. The closed ball in X with center x and radius r is denoted B@, r), but for the unit ball we write XI=B(O, 1). The dual space of X is written X*. The convex hull of a set S is written cony (S) and the set of extreme points

Bounded approximation properties via integral and nuclear operators

Published version of an article in the journal:Proceedings of the American Mathematical Society. Also available from the publisher, Open Access

Complex Banach spaces whose duals areL1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:i)A* is isometric to anL1(μ)-space;ii)every family of 4 balls inA with the weak intersection property has a non-empty intersection;iii)every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

Hahn-Banach extension operators and spaces of operators

Let X C Y be Banach spaces and let A C B be closed operator ideals. Let Z be a Banach space having the Radon-Nikodym property. The main results are as follows. If Φ: A(Z,X)* → B(Z, Y)* is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators Φ i : X* → Y*, i ∈ I, such that Z = Σ i ○+ I ○+ 1 Z ΦΦi , where Z ΦΦi = {z ∈ Z: Φ(x* ⊗ z) = (Φ i x*) ⊗ z,?x* ∈ X*}. If A(Z, X) is an ideal in B(Z, Y) for all equivalently renormed versions Z of Z, then there exist Hahn-Banach extension operators Φ: A(Z, X)* → B(Z, Y)* and Φ: X* → Y* such that Z = Z ΦΦ .

Centers of symmetry in finite intersections of balls in Banach spaces

It is proved that a real Banach spaceX is aG-space (Cσ - space)if and only if the non-empty intersection of three balls with equal radii (any three balls) has a center of symmetry.