# Recent Advances in Banach lattices (18w5087)

Arriving in Oaxaca, Mexico Sunday, April 29 and departing Friday May 4, 2018

## Organizers

Anthony Wickstead (Queen's University Belfast)

Ben de Pafter (Delft University of Technology)

Gerard Buskes (University of Mississippi)

Vladimir Troitsky (University of Alberta)

## Objectives

In this workshop, we consider recent developments in the area of Banach lattices. The goal of the workshop is to bring together the leading experts and active young researchers to discuss the current and future directions of these developments and to identify potential applications and the main open problems. We also plan to understand the "big picture" of connections between these developments and other areas of Functional Analysis.

We will focus on the following topics.

1. Martingales in vector and Banach lattices. Historically, martingales have been modelled using tools of Probability Theory. Around 2005, a new approach was started in a series of papers by J.Grobler, C.Labuschagne, V. Troitsky, B. Watson, and others, in which martingales are modelled using tools of vector or Banach lattices. In this approach, a filtration is a sequence of positive projection on the lattice, while a martingale is a sequence of vectors in the lattice, which is adapted to the filtration. This approach generalizes the classical approach; it is sometimes called "measure-free martingale theory". Many facts of the classical martingale theory have been "migrated" into the "measure-free" setting. In particular, the theory of sctochastic integration has recently been extended to the measure-free setting by Grobler and Labuschagne, and Doob's a.e. convergence theorem has been extended by by Gao and Xanthos using the technique of uo-convergence. There is a considerable interplay going on between the "measure-free" approach and the classical theory of stochastic processes. It combines various techniques and facts of Probability Theory with power of Functional Analysis.

Goals. (1.1) Discuss what other parts of the classical theory of stochastic processes would extend to the measure-free setting; e.g., Burkholder-type martingale inequalities. (1.2) Understand when the new theory reduces to the classical theory (e.g., using uo-convergence) and when the results of the classical theory can be applied in the new settings. (1.3) Understand the connections between the Banach lattice and the vector lattice variants of the theory; ideally, we need a "unified" theory. (1.4) Study the connections between measure-free martingales in Banach lattices and classical Banach-space-valued martingales.

2. Tensor products of Banach lattices and Polynomials on Banach lattices. In 2006, Y.Benyamini, S.Lassalle, and J.G.Llavona revived the topic of homogeneous orthogonally additive polynomials on Banach lattices, which had originally been started by Sundaresan in 1991. They connected the study of such polynomials with concavifications and this has subsequently led to a very rapid development. Newer developments have used the theory of Fremlin tensor products, thus establishing the link with an already well-established theory of homogeneous polynomials and multilinear maps on Banach spaces. Applications to C*-algebras, Fourier analysis, and matrix analysis are appearing as well. A potential development of complex analysis on Banach lattices is to be expected.

Goals: (2.1) Connect Banach lattices with complex analysis on infinite dimensional spaces. (2.2) Connect and contrast the new theory with the older theory of regular operators on Banach lattices.

3. Geometry and unconditional structures: Unconditionality has been a fruitful tool in Banach space theory showing many interactions with combinatorics, logic and other fields of analysis, such as greedy approximation, for instance. Since the celebrated paper by T.Gowers and B.Maurey, Banach spaces without unconditional basic sequences are known to exist. These examples have led to the theory of Hereditarily Indecomposable spaces developed by S.Argyros, R.Haydon, T.Schlumprecht. On the other hand, several notions related to unconditionality have been introduced in the recent years. These include partial unconditionality, Schreier unconditionality, random unconditionality, and many structural results have been obtained with them. Moreover, these notions go deeper into the relation between Banach space theory and combinatorics, set theory and probability.

From the point of view of Banach lattice theory, spaces with unconditional basis can be seen as atomic Banach lattices. Moreover, Banach lattices have plenty of unconditional sequences, in particular every sequence of disjoint vectors. In the recent years, there has been a renewed interest in the geometric properties of disjoint sequences in Banach lattices since the introduction of disjointly homogeneous spaces by J.Flores, P.Tradacete and V.Troitsky. Furthermore, the understanding of these properties provides information on the operators between Banach lattices.

Goals: (3.1) Determine whether random unconditionality is present in every Banach space. (3.2) Understand the behavior of the constants involved in partial unconditionality and its combinatorial counterparts. (3.3) Explore the implications of Ramsey theory for Banach lattices.

4. Multinorms. Theory of multinorms was started in the 2012 manuscript by G.Dales and M.Polyakov motivated by problems in Banach Algebras, followed by papers by M.Daws, H.L.Pham, P.Ramsden, and O.Blasco. It turned out that this new theory has strong connections to several classical areas of Functional Analysis, including tensor products of Banach spaces, absolutely summing operators, and Banach lattices. It follows from results of G.Pisier, P.Cassaza, and P.Nielsen about tensor products that every multinormed space can be represented as a subspace of a Banach lattice. It also follows from results of G.Pisier and L.MacClaran that an operator on a Banach lattice is multibounded with respect to the canonical multinorm iff its adjoint is regular. Many of these results can be generalized to $p$-multinorms for $1\le p\le\infty$. There are also connections between multinorms on $X$ and certain tensor norms on $c_0\otimes X$ and, more generally, between $p$-multinorms on $X$ and certain tensor norms on $\ell_p\otimes X$.

Goals: (4.1) Find Banach lattice representations for specific multinorms developed by Dales, Polyakov, et al. (4.2) Extend the representation theorems mentioned above to general $p$-multinorms. (4.3) Characterize $p$-multibounded operators between Banach lattices.

Many of these topics and goals will be accessible to graduate students. The workshop will expose students and young researches to actively developing areas and connections between them, as well as to many open problems.

We will focus on the following topics.

1. Martingales in vector and Banach lattices. Historically, martingales have been modelled using tools of Probability Theory. Around 2005, a new approach was started in a series of papers by J.Grobler, C.Labuschagne, V. Troitsky, B. Watson, and others, in which martingales are modelled using tools of vector or Banach lattices. In this approach, a filtration is a sequence of positive projection on the lattice, while a martingale is a sequence of vectors in the lattice, which is adapted to the filtration. This approach generalizes the classical approach; it is sometimes called "measure-free martingale theory". Many facts of the classical martingale theory have been "migrated" into the "measure-free" setting. In particular, the theory of sctochastic integration has recently been extended to the measure-free setting by Grobler and Labuschagne, and Doob's a.e. convergence theorem has been extended by by Gao and Xanthos using the technique of uo-convergence. There is a considerable interplay going on between the "measure-free" approach and the classical theory of stochastic processes. It combines various techniques and facts of Probability Theory with power of Functional Analysis.

Goals. (1.1) Discuss what other parts of the classical theory of stochastic processes would extend to the measure-free setting; e.g., Burkholder-type martingale inequalities. (1.2) Understand when the new theory reduces to the classical theory (e.g., using uo-convergence) and when the results of the classical theory can be applied in the new settings. (1.3) Understand the connections between the Banach lattice and the vector lattice variants of the theory; ideally, we need a "unified" theory. (1.4) Study the connections between measure-free martingales in Banach lattices and classical Banach-space-valued martingales.

2. Tensor products of Banach lattices and Polynomials on Banach lattices. In 2006, Y.Benyamini, S.Lassalle, and J.G.Llavona revived the topic of homogeneous orthogonally additive polynomials on Banach lattices, which had originally been started by Sundaresan in 1991. They connected the study of such polynomials with concavifications and this has subsequently led to a very rapid development. Newer developments have used the theory of Fremlin tensor products, thus establishing the link with an already well-established theory of homogeneous polynomials and multilinear maps on Banach spaces. Applications to C*-algebras, Fourier analysis, and matrix analysis are appearing as well. A potential development of complex analysis on Banach lattices is to be expected.

Goals: (2.1) Connect Banach lattices with complex analysis on infinite dimensional spaces. (2.2) Connect and contrast the new theory with the older theory of regular operators on Banach lattices.

3. Geometry and unconditional structures: Unconditionality has been a fruitful tool in Banach space theory showing many interactions with combinatorics, logic and other fields of analysis, such as greedy approximation, for instance. Since the celebrated paper by T.Gowers and B.Maurey, Banach spaces without unconditional basic sequences are known to exist. These examples have led to the theory of Hereditarily Indecomposable spaces developed by S.Argyros, R.Haydon, T.Schlumprecht. On the other hand, several notions related to unconditionality have been introduced in the recent years. These include partial unconditionality, Schreier unconditionality, random unconditionality, and many structural results have been obtained with them. Moreover, these notions go deeper into the relation between Banach space theory and combinatorics, set theory and probability.

From the point of view of Banach lattice theory, spaces with unconditional basis can be seen as atomic Banach lattices. Moreover, Banach lattices have plenty of unconditional sequences, in particular every sequence of disjoint vectors. In the recent years, there has been a renewed interest in the geometric properties of disjoint sequences in Banach lattices since the introduction of disjointly homogeneous spaces by J.Flores, P.Tradacete and V.Troitsky. Furthermore, the understanding of these properties provides information on the operators between Banach lattices.

Goals: (3.1) Determine whether random unconditionality is present in every Banach space. (3.2) Understand the behavior of the constants involved in partial unconditionality and its combinatorial counterparts. (3.3) Explore the implications of Ramsey theory for Banach lattices.

4. Multinorms. Theory of multinorms was started in the 2012 manuscript by G.Dales and M.Polyakov motivated by problems in Banach Algebras, followed by papers by M.Daws, H.L.Pham, P.Ramsden, and O.Blasco. It turned out that this new theory has strong connections to several classical areas of Functional Analysis, including tensor products of Banach spaces, absolutely summing operators, and Banach lattices. It follows from results of G.Pisier, P.Cassaza, and P.Nielsen about tensor products that every multinormed space can be represented as a subspace of a Banach lattice. It also follows from results of G.Pisier and L.MacClaran that an operator on a Banach lattice is multibounded with respect to the canonical multinorm iff its adjoint is regular. Many of these results can be generalized to $p$-multinorms for $1\le p\le\infty$. There are also connections between multinorms on $X$ and certain tensor norms on $c_0\otimes X$ and, more generally, between $p$-multinorms on $X$ and certain tensor norms on $\ell_p\otimes X$.

Goals: (4.1) Find Banach lattice representations for specific multinorms developed by Dales, Polyakov, et al. (4.2) Extend the representation theorems mentioned above to general $p$-multinorms. (4.3) Characterize $p$-multibounded operators between Banach lattices.

Many of these topics and goals will be accessible to graduate students. The workshop will expose students and young researches to actively developing areas and connections between them, as well as to many open problems.