
Banach spaces of continuous and Lipschitz functions (GF2022230L)
from 01/01/2020
to 31/12/2022 investigator

Objectives:
The supreme goal of this project is to study Banach spaces of continuous and Lipschitz functions over compact spaces from the points of view of various branches of mathematics. Since every Banach space can be isometrically embedded into a Banach space of continuous functions on a compact space, such spaces constitute one of the most important classes of Banach spaces and thus their better understanding will lead to sourcing broader knowledge and comprehension of all Banach spaces in general. We propose to apply mathematical techniques and methods originating from different areas of modern analysis, geometry, topology, and logic. Such a universal approach will allow us to obtain more profound insight into the structure of Banach spaces and related objects, aiming at solving several open problems in this area. Expected results will offer new tools for studying and classifying the topology and geometry of spaces of continuous and Lipschitz functions, as well as the corresponding compact and metric spaces.



Topological and geometrical properties of Banach spaces and operator algebras II (GA1700941S)
from 01/01/2017
to 31/12/2019 main investigator

Objectives:
We wish to investigate the structure of Banach spaces, C*algebras and Jordan algebras and their relationship. Main topics include quantitative approach to Banach spaces, various methods of separable reduction, decompositions of Banach spaces to smaller subspaces, integral representation of affine Baire functions, descriptive properties of weak topologies, small sets in Banach spaces and Polish groups, universal spaces in various categories of Banach spaces, operators and their numerical ranges, structure of abelian subalgebras of a C*algebra, of associative subalgebras of a Jordan algebra and related structures, different types of order in operator algebras, representation of morphisms on various substructures of operator algebras, Bell's inequalities and quantum correlations. We wish to focus especially on those problems where the mentioned areas intersect each other and by solving them to contribute to clarification of connections among various areas of functional analysis.



Topological and geometrical properties of Banach spaces and operator algebras (GAP201/12/0290)
from 01/01/2012
to 31/12/2016 main investigator

Objectives:
We would like to study the structure of Banach spaces, spaces of continuous functions, C* algebras and their relationship. Main topics will include quantitative properties of Banach spaces, decompositions of Banach spaces to smaller subspaces, descriptive properties of the weak topology, James' boundaries of compact convex sets, Baire classes of strongly ane functions, noncommutative Choquet theory, weakly compact sets and spaces they generate, problems of BanachStone type for spaces of continuous functions, uniformly continuous functions and ane functions, various types of universal Banach spaces, existence of fixed points and approximate fixed points, noncommutative measure theory, structures on the set of all abelian subalgebras of a C* algebra and a Jordan algebra, representations of operator algebras using weights and completely positive maps, new types of orders on operators. We would like to pay special attention to mutual influence of particular structures and to subsequent connecting of dierent areas of functional analysis and clarifying their mutual relationships.



Nonlinear analysis in Banach spaces (7AMB12FR003)
from 01/01/2012
to 31/12/2013 investigator

Programme type: MOBILITY  France
Objectives:
The goal of this project is to contribute to a better understanding of the following topics:



Variational inequalities in equilibrial optimization (MEB021024)
from 01/01/2010
to 31/12/2011 main investigator

Programme type: MOBILITY  France
Objectives:
Development of tools of variational analysis suitable for study of local properties of solutions of variational inequalities.
Analysis of the stability and sensibility of multivalued mappings which assign suitable sets of solutions to the data of problems or to perturbing parameters.
Application of obtained results on selected equilibrial problems of continuum mechanics and of mathematical economy.



Topological and geometrical structures in Banach spaces (IAA100190901)
from 01/01/2009
to 31/12/2011 main investigator

Stability of weak Asplund spaces and Gateaux differentiability spaces, Banach spaces with projectional skeleton, stability of Valdivia compacta, topological characterizations of some classes of Banach spaces, compact convex sets  duality of compact spaces and Banach spaces and topological classification, weak topology of weakly Kanalytic Banach spaces, geometric properties of boundaries of compact convex sets, strongly affine functions and Baire classes of Banach spaces, isomorphic properties of L_1preduals, descriptive properties of ranges of derivatives on Banach spaces, descriptive properties of norm and weak topologies on Banach spaces, binormality in Banach spaces, spaces of continuous functions, renormings and differentiability, weakstar uniformly KadecKlee norms, Lipschitz homeomorphisms, Markushevich bases, extensions of special Lipschitz mappings, retracts, absolutely minimal Lipschitz functions, Gurarij space, compacta with extra structure.



Infinite dimensional analysis (201/07/0394)
from 01/01/2007
to 31/12/2009 investigator

We propose to work on problems in several areas of infinite dimensional analysis. Regarding the structure of Banach spaces, does every Banach space contain a monotone basic sequence? What is the optimal value of boundedness constant for an Mbasis of a general Banach space. We conjecture the answer is 2. What are the structural consequences of a long Schauder basis. In connection with renorming theory, find an example of a LUR renormable space with Mbasis, but no strong Mbasis. Clarify the relationship between the separable complementation property and PRI. Clarify the relationship between Szlenk index and dentability index, for which there exists a nonconstructive proof of dependence. Try to generalize the Szlenk index technique in the context of topologies framented by a metric and beyond. In particular, as an application, does there exist a universal Corson compact space?



Geometry of weakly Lindelöf determined spaces (IAA100190610)
from 01/01/2006
to 31/12/2008 main investigator

Renorming theorems in weakly Lindelöf determined Banach spaces for the so called uniformly KadecKlee smoothness and its variants. The use of Shlenk and dentability index for these renormings. Study of l_p generated Banach spaces for p>2. For a given Banach space with some geometrical property, finding a space with the same property and moreover having an unconditional basis, which is densely embedded in the original space. Investigation of the geometry of Banach spaces possessing an unconditional basis in the spirit of our recent results. Study of long Schauder bases.



The structure of Banach spaces (IAA100190502)
from 01/01/2005
to 31/12/2007 investigator

The aim of the project is to contribute to the solution of some of the fundamental question regarding the structure of Banach spaces. In the contextof separable spaces it is especially the containment of copies of cO into quotients of polyhedral spaces. The question on the existence of smooth separating functions on asplund spaces. Problems of the density of smooth renormings and the existence of lipschitz retractions for certain C(K)spaces. The boundary problem, ie. generalization of Rainwaters theorem to the case of arbitrary boundary and dropping the sequentiality assumption. Structure of biorthogonal systems and Markushevich basas in nonseparable Banach spaces.
