On the group invariant Ekeland's variational principle and its equivalences.

During the years, the group invariant mappings have been mainly studied in measure theory. Recently, this studies have extended to other fields including functional analysis. In this talk we will present a group invariant version of the well-known Ekeland's variational principle (EVP). To do so we will briefly present the three basic notions of group invariant and convexity with respect to the group. Then we will present some of the widely extended applications of the EVP, all of them in their respective group invariant version, such as the Palais-Smale minimizing sequences, the Bronsted-Rockafellar theorem, the Bishop-Phelps theorem, and give a full description of the linear and continuous group invariant functionals space. Finally we will present some equivalences of the Ekeland variational principle to the petal and the drop theorem, in their respective group invariant versions. This is joint work with Javier Falcó.

1. Ekeland I., On the variational principle, Journal mathematical analysis and applications 47.2, (1974), pp. 324-353.
2. Fabian M.; Habala P.; H\'ajek P.; Montesinos V.; Zizler V., Banach space theory: the basis for linear and nonlinear analysis, Springer, 2011.
3. Falcó J.; Isert D., Group invariant variational principles, ArXiv (2023).
4. Penot J.-P., The drop theorem, the petal theorem and Ekeland's variational principle, Nonlinear analysis theory, methods and applications, 10 (1986), pp. 813--822.
5. Oettli, W. and Th\'era, M., Equivalents of Ekeland's variational principle, Bulletin of the Austrlian Mathematical Society, 48 (1993), pp. 385--392.

Norm-Attaining and Non-Norm-Attaining Group-Invariant Operators and multilinear forms

In this talk we present some recent results in the theory of norm-attaining operators within the framework of group invariant mappings—those satisfying $T(x) = T(g(x))$ for every operator $G \subset L(X)$, where $G$ represents a group of linear operators from $X$ to $X$. Within this context, we delve into geometric properties such as group invariant versions of the Hahn-Banach theorem and extend classical results of the norm-attaining theory. Furthermore, we will also present several lineability properties of the set of non-norm-attaining mappings in this framework.

The disorder of the recent counterexample to the Complemented Subspace Problem

The well-known Complemented Subspace Problem for C(K)-spaces, which asks whether every complemented subspace of a C(K)-space is also isomorphic to a C(K)-space, has been recently solved by G. Plebanek and A. Salguero-Alarcón in the negative.

The goal of this talk is to sketch the main ideas behind their construction and to reformulate it in terms of the order, showing that their space not only fails to be isomorphic to a C(K)-space but it is not even isomorphic to a Banach lattice.

This talk is based on a joint work with David de Hevia, Alberto Salguero-Alarcón and Pedro Tradacete.

On $\omega$-Corson and $NY$ compact spaces

Given a family $\{X_\gamma:\gamma\in \Gamma\}$ of topological spaces $X_\gamma$ and a point $a=(a_\gamma)_{\gamma\in \Gamma}\in X=\prod_{\gamma\in \Gamma} X_\gamma$, we define the $\sigma$-product in $X$ centered at $a$ as $$\sigma(X,a)=\{x\in \prod_{\gamma\in \Gamma} X_\gamma:\{\gamma\in \Gamma:x(\gamma)\neq a_\gamma\}<\omega\}.$$

A compact space $K$ is called $\omega$-Corson (resp. $NY$ compact) if $K$ embeds into a $\sigma$-product of intervals (resp. a $\sigma$-product of compact metric spaces). Clearly, every $\omega$-Corson compactum is $NY$ compact and every $NY$ compact space is Eberlein compact. In my talk I will discuss some recent results about $\omega$-Corson and $NY$ compact spaces, as well as related classes of $\omega$-Valdivia and $NY$-Valdivia compacta. Joint work with Antonio Avilés.

Representation of maxitive functionals on lattices

A functional on a lattice is maxitive if it commutes with the supremum operation. Motivated by questions in risk analysis, we provide a representation result for maxitive functionals on Dedekind complete lattices.

The key assumption is that the partial order admits a numerical representation which depends lower-semicontinuously on a parameter. The result is illustrated by discussing the representation for different stochastic orderings of random variables.

This talk is based on joint work with Michael Kupper.

Espacios Lipschitz-free. Integrales de moléculas.

Los espacios Lipschitz-free son linealizaciones canónicas de espacios métricos que se obtienen como espacios de funcionales sobre espacios de funciones Lipschitz. Sus elementos básicos son las llamadas moléculas, que representan cocientes incrementales. Todo elemento de un espacio Lipschitz-free puede obtenerse como una serie casi convexa de moléculas. En esta línea de investigación, conjunta con E. Pernecká (Praga) y R. J. Smith (Dublin), estudiamos qué elementos pueden obtenerse como una serie, o más en general una integral, convexa de moléculas. Mostraré los resultados obtenidos hasta ahora, y su relación con otros problemas abiertos como el de determinar los puntos extremos de la bola o la densidad de las funciones Lipschitz que alcanzan la norma fuertemente.

Asymptotic smoothness in Banach spaces, Three-space properties and applications.

Four linear properties, related to the asymptotic smoothness, will be considered in this talk: $\mathsf{T}_p$, $\mathsf{A}_p$, $\mathsf{N}_p$ and $\mathsf{P}_p$. We will be interested in their stability under some nonlinear equivalences and if they are three-space properties or not. After recalling the already known results, we will complety solve the three-space problem for this family of properties and derive from characterizations of $\mathsf{A}_p$ and $\mathsf{N}_p$ in terms of equivalent renormings, new coarse Lipschitz rigidity results for these classes. This is part of a joint work with R. Causey and G. Lancien.

M-ideals, yet again: the case of real JB*-triples.

We prove that a subspace of a real JBW*-triple is an M-summand if and only if it is a weak*-closed triple ideal. As a consequence, M-ideals of real JB*-triples correspond to norm-closed triple ideals. This is a newfangled treatment of the classical notion of M-ideal in the real setting by a fully new approach due to the unfeasibility of the known arguments in the setting of complex C*-algebras and JB*-triples. The results in this note also provide a full characterization of all M-ideals in real C*-algebras, real JB*-algebras and real TROs.

Radius and diameter of slices in Banach spaces.

In 2006, Y. Ivakhno considered two different properties which, somehow, encode the fact that slices of the unit ball of a given Banach space are big. On the one hand, we say that a Banach space $X$ has the slice-D2P if $$d(X):=\inf\{diam(S): S\mbox{ is a slice of }B_X\}=2.$$

On the other hand, we say that $X$ has the r-BSP if, for every slice $S$ of $B_X$ we have, $$r(S):=\inf\{r>0: S\subseteq B(x,r)\mbox{ for some }x\in X\}\geq 1.$$ Similarly to $d(X)$, we can define $$r(X):=\inf\{r(S): S\mbox{ is a slice of }B_X\}.$$ Y. Ivakhno proved in [2] that $d(X)=2$ implies $r(X)\geq 1$. The converse was left as an open question. Based on the research developed in [1,3], the aim of this talk is to show that this converse does not hold. Moreover, we will explore how small $d(X)$ can be if $r(X)\geq 1$.

This work was supported by MCIN/AEI/10.13039/501100011033: Grant PID2021-122126NB-C31, Junta de Andalucía: Grant FQM-0185, by Fundación Séneca: ACyT Región de Murcia grant 21955/PI/22 and by Generalitat Valenciana project CIGE/2022/97.

References:

• R. Haller, J. Langemets, V. Lima, R. Nadel and A. Rueda Zoca, On Daugavet indices of thickness, J. Funct. Anal. 280 (2021), article 108846.
• Y. Ivakhno, On sets with extremely big slices, J. Math. Phys. Anal. Geom. 2 (2006), no. 1, 94--103.
• A. Rueda Zoca, Diameter, radius and Daugavet index thickness of slices in Banach spaces, accepted in Isr. J. Math.

Lie--Trotter formulae in Jordan-Banach algebras with applications to the study of spectral-valued multiplicative functionals.

We establish some Lie--Trotter formulae for unital complex Jordan-Banach algebras, showing that for each couple of elements $a,b$ in a unital complex Jordan-Banach algebra $\mathfrak{A}$ the identities $$ \lim_{n\to \infty} \left(e^{\frac{a}{n}}\circ e^{\frac{b}{n}} \right)^{n} = e^{a+b},\ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{2 a+b}, \hbox{ and }$$ $$ \lim_{n\to \infty} \left(U_{e^{\frac{a}{n}},e^{\frac{c}{n}}} \left( e^{\frac{b}{n}}\right) \right)^{n} = e^{a+b + c}$$ hold.

These formulae are employed in the study of spectral-valued (non-necessarily linear) functionals $f:\mathfrak{A}\to \mathbb{C}$ satisfying $f(U_x (y))=U_{f(x)}f(y),$ for all $x,y\in \mathfrak{A}$. We prove that for any such a functional $f,$ there exists a unique continuous (Jordan-)multiplicative linear functional $\psi\colon \mathfrak{A}\to\mathbb{C}$ such that $ f(x)=\psi(x),$ for every $x$ in the connected component of set of all invertible elements of $\mathfrak{A}$ containing the unit element. If we additionally assume that $\mathfrak{A}$ is a JB$^*$-algebra and $f$ is continuous, then $f$ is a linear multiplicative functional on $\mathfrak{A}$.

The new conclusions are appropriate Jordan versions of results by Maouche, Brits, Mabrouk, Shulz, and Touré.