CONFERENCES

Antonio Avilés

(University of Murcia)

Compact spaces associated to separable Banach lattices

Given a vector x of a Banach lattice X, the ideal generated by x is canonically identified with a C(K) space. The compact spaces that arise in this way from Banach lattices constitutes an intriguing class that we will analyse in our talk.

Jesús M. F. Castillo

(University of Extremadura)

Invitation to the Club of People Who Stare at Diagrams

This is a personal invitation to the Club of People Who Stare at Diagrams.

Willian H. G. Corrêa

(University of São Paulo)

Interpolation of Orlicz sequence spaces and the C[0,1]-extension property

Androulakis, Kalton and Cazaku showed how to use Fenchel-Orlicz spaces to obtain twisted sums of Orlicz spaces. That is, given a suitable Orlicz space 𝓁φ and a non negative Lipschitz function on the real line they obtain a Fenchel-Orlicz space 𝓁ϕ which contains 𝓁φ as subspace with respective quotient isomorphic to 𝓁φ. One consequence is that every operator from 𝓁2 into C[0,1] extends to the Kalton-Peck space. In this talk we will discuss the connection of those results with the theory of complex interpolation of Banach spaces.

Valentin Ferenczi

(University of São Paulo)

A solution to a problem of Pietsch on operator ideals

An operator ideal J in the sense of Pietsch is proper when Id_X belongs to J only if the space X is finite dimensional. Pietsch (1979) asked whether there is a largest ideal among proper ideals, and conjectured that the ideal of inessential operators is such an object. However, using works of Gowers-Maurey (1997) and Aiena-González (2000), we shall answer the question by proving that there is no largest proper ideal. To conclude we shall indicate direC(K)tions in which this result could be extended.

Gilles Godefroy

(CNRS)

Weak-star Borel subspaces of 𝓁∞

The space 𝓁∞ of bounded real sequences is an important example of non-separable C(K)-space, and its weak-star topology is somewhat canonical. We investigate the norm-closed linear subspaces of 𝓁∞ which are weak-star Borel. The operator of anti-differentiation is used to show that such spaces exist which are of arbitrarily high Borel class. Moreover, we investigate if a weak-star Borel subspace of 𝓁∞ which is isomorphic to 𝓁∞ is necessarily weak-closed, and we prove that it is indeed so in a number of cases.

David de Hevia

(Institute of Mathematical Sciences - ICMAT)

A counterexample to the complemented subspace problem in Banach lattices

The Complemented Subspace Problem in Banach lattices asks whether any complemented subspace of a Banach lattice must be isomorphic to a Banach lattice. In 2021, G. Plebanek and A. Salguero-Alarcón provided an example, denoted by PS2 , of a one-complemented subspace of a C(K)-space which is not isomorphic to any C(L) (with L being an arbitrary compact Hausdorff space). This answered in the negative the Complemented Subspace Problem in the C(K) setting. We shall show that, in fact, the space PS2 is not even isomorphic to a Banach lattice.

This talk is based on a joint work with Gonzalo Martínez-Cervantes, Alberto Salguero-Alarcón and Pedro Tradacete.

Tomasz Kania

(Czech Academy of Sciences in Prague)

Complex C(K)-spaces of topological stable rank 1

We investigate the problem of when a unital Banach algebra has (uniformly) open multiplication. For (complex) C(K)-spaces, we show that this is precisely so when the covering dimension of K is at most one. We find sufficient conditions for a commutative Banach *-algebra to have open multiplication.

Antonio Martínez-Abejón

(University of Oviedo)

On the subspace structure of L1: the strong property (B)

An infinite dimensional Banach space X is said to possess the property (B) if for every infinite dimensional, closed subspace Y there is a decomposition X=X1 ⊕ X2 such that the subspaces Yi := Y ∩ Xi (i=1,2) are both infinite dimensional. In addition, if Y1, Y2 and X are isomorphic then X is said to possess the strong property (B).

The property (B) was introduced by Cross, Ostrowskii and Shevchik in 1994 [3] in order to translate to the Banach space setting a theorem about operator ranges on Hilbert spaces due to Dixmier. Cross and his coauthors found two natural classes of Banach spaces with property (B):

• spaces with unconditional basis;

• subprojective spaces.

In particular, they proved that Lp(0,1 has property (B) if 1<p<∞, does not have if p=∞, and left as an open question the case p=1.

The strong property (B) was introduced in 2005 by Martínez-Abejón, Odell and Popov with the aim of digging into the structure of complemented subspaces of L1(0,1, which is not yet well understood.

The subject of my talk will deal with the recent discovery that Lp(0,1) has the strong property (B) for all 1<p<∞. The proofs for the cases 1<p<∞ and p=1 are certainly different: while the cases 1<p<∞ are based upon the unconditionality of the Haar basis, the case p=1 is remarkably more difficult and relies on Aldous' theorem that shows that L1(0,1) contains copies of 𝓁p for all 1<p≤2, a result by Dacunha-Castelle and Krivine on the isometrically asymptotic copies of 𝓁p in L1(0,1) and a change of density due to Berkes and Rosenthal.

References

[1] D.J. Aldous. Subspaces of L1, via random measures. Trans. Amer. Math. Soc. 267, n. 2 (1981), 445–463.

[2] I. Berkes, H.P. Rosenthal. Almost exchangeable sequences of random variables. Z. Wahrsch. Verw. Gebiete 70 (1985), 473–507.

[3] R.W. Cross, M.I. Ostrovskii, V.V. Shevchik. Operator Ranges in Banach spaces. I. Math. Nachrichten 173 (1995), 91114.

[4] D. Dacunha-Castelle, J.L. Krivine. Sous-espaces de L1. Israel J. Math. 26, n.3-4 (1977), 320–351.

[5] J. Dixmier. Étude sur le Varietés et les Opérateurs de Julia. J. Math. Pures Appl. 28 (1949), 321–358

[6] A. Mart ́ınez-Abejón. The strong property (B) for Lp spaces. Mediterr. J. Math. 19, 5 (2022).

[7] A. Martínez-Abejón, M.M. Popov, E. Odell. Some open problems on the classical function space L1. Matematychnii Studii, vol. 24, n.2, (2005), 173-191.

Gonzalo Martínez-Cervantes

(University of Alicante)

Results and questions on sequential properties in dual balls with the weak*-topology

A topological space is said to be Fréchet-Urysohn (FU) if the closure of every subspace coincides with the set of limits of sequences in the subspace. A generalization of FU property are sequentiality and countable tightness. A topological space is sequential if every nonclosed subspace contains a sequence converging to a point which is not in the subspace. On the other hand, a topological space has countable tightness if the closure of every subspace coincides with the union of closures of countable subsets of the subspace.

During this talk we shall study these and convex analogues of these properties in dual balls with the weak*-topology. In particular, we shall focus on the following two questions:

1) Are these properties three-space properties?

2) What is the relation among these properties?

Pedro Tradacete

(Institute of Mathematical Sciences - ICMAT)

Diagrams for Banach lattices

The free Banach lattice generated by a Banach space E is a Banach lattice FBL[E] together with a linear isometric embedding 𝛿: E → FBL[E] with the following universal property: For every Banach lattice X and every bounded linear map T: E → X there is a unique linear lattice homomorphism Ť: FBL[E] → X such that Ť o 𝛿 = T. We will present some recent results on free Banach lattices, exhibiting the interaction between Banach spaces and Banach lattices.