Final Report Summary - LOGICANDBANACHSPACES (Set theory, topology and Banach spaces)
The general objective of the project was to work in the border line of set theory, topology and Banach spaces in such topics as infinite-dimensional topology or spaces of continuous functions. In particular we hoped to be able to use the advanced techniques in set theory that the researcher had learnt previously in some problems in the field of functional analysis. We had some concrete sets of problems as an initial guideline for our research: these included problems about:
(1) operator equations;
(2) homological methods in Banach spaces;
(3) renorming theory of Banach spaces;
(4) infinite-dimensional topology; and
(5) spaces of continuous functions on compact spaces.
The list of topics within the field where set theory, functional analysis and topology meet, and we could expect to make some progress, was expected to be expanded by the interaction with different experts.
The research activity can be organised along the following lines:
Line 1 (collaboration with G. Plebanek and J. Rodriguez)
After arriving in Murcia, the researcher initiated a collaboration with Jose Rodriguez from this university. He was working in a number of problems on functional analysis that seemed to have a set-theoretic flavor, and seemed thus to suit perfectly in the spirit of the project. One of our first decisions was to invite Grzegorz Plebanek to visit us and to discuss these problems, because, being an expert on abstract measure theory with a knowledge on Banach spaces and set-theory, he had a suitable profile that for the purposes of the problems we wanted to study. This was quite a success, and at the end we have four papers authored by the team Aviles-Plebanek-Rodriguez contributing to the success of this project. We briefly describe each of them.
1. 'The McShane integral in weakly compactly generated spaces' (J. Funct. Anal.)
In this paper, a problem posed by L. Di Piazza and D. Preiss is solved by constructing a function f:[0,1] --> X which is Pettis-integrable but not McShane integrable, yet X is a weakly compactly generated Banach space.
2. 'Measurability in C(2^k) and Kunen cardinals' (Israel J. Math.)
In this paper, a substantial improvent of a result of Fremlin is made, by showing that in the space C(K) of continuous functions on the generalised Cantor cube K = 2^k, the Baire and the Borel sigma-algebras coincide if and only if k is a Kunen cardinal.
3. 'A weak- separable C(K)- whose ball is not weak- separable' (Trans. A.M.S.)
In this paper, a space C(K) is constructed with the properties stated in the title and some measurability questions on it are also discussed. The only previously known example of such a space had been constructed by Talagrand assuming the continuum hypothesis, but this space exists in ZFC.
4. 'On Baire measurability in spaces of continuous functions' (J. Math. Anal. App.)
In this paper, under CH, a compact space K is constructed which has a probability measure which is in the w*-sequential closure of the convex hull of Dirac measures, but it is not the w*-limit a sequence of them.
Line 2 (collaboration with P. Koszmider)
One important open question of set-theoretic topological nature in nonseparable Banach space theory was whether the continuous image of a Radon-Nikodym compact space is again Radon-Nikodym compact. The researcher had worked on this topic in his PhD, and Piotr Koszmider from Warsaw got interested in the problem and in this previous work on it and he had some ideas to use the technique of resolutions of compact spaces to make some progress. A collaboration started, P. Koszmider visited Murcia on the fall of 2011 and a counterexample to the problem was finally obtained that has been recently accepted for publication in Duke Math. J. The result has important consequences in spaces of continuous functions, as it provides a space C(K) which is not isomorphic to C(L) with L zero-dimensional, despite the fact that K is 'very nice' (it has, for instance, lot of convergent sequences).
There is another joint work with P. Koszmider, in which, answering a question by Haily, Kaidi and Rodriguez-Palacios, a Banach space X is constructed, actually of the form C(K), such that every injective operator on X is an isomorphism.
Line 3 (collaboration with S. Todorcevic)
The researcher has been engaged with Stevo Todorcevic in developing the theory of multiple gaps, a topic on which they have written three papers during this period (one in Fund. Math, another in Combinatorica and the third is currently submitted for publication). The notion of multiple gap is a purely set-theoretic one, constituting a high dimensional generalisation of the classical concept of gap tracing back to Hausdorff. Aviles and Todorcevic invented this as a tool to deal with injectivity in Banach spaces of continuous functions, in the context of what we call 'homological methods in Banach spaces', in connection with some results of Ditor. However, it was realised later that there is a deep structural theory of these objects that has an interest on its own and quite different applications. This is particularly true in the case of analytic multiple gaps, to which the second and third paper are exclusively devoted. This analytic theory has found different applications in separable Banach space theory, as it allows to identify the canonical ways in which different basic sequences can be mixed in a single sequence. Part of this research was done during a visit of Antonio Aviles to Paris in the summer of 2011.
Line 4 (homological methods)
There have been actually some progress on the topic of 'homological methods in Banach spaces'. Christina Brech from Sao Paulo was invited to visit in Murcia, and this lead to a discussion about the results on the paper 'on Banach spaces of universal disposition' by Aviles et al. It was realised that, though the general language of category theory, one could connect problems in Banach space theory with problems on set theory considered in a paper of Dow and Hart. Putting all these ideas together, it was possible to answer some questions from both sides, and this gave origin to the paper 'A Boolean algebra and a Banach space obtained by push-out iteration' by Aviles and Brech. Later, Wieslaw Kubis from Prague, visited Murcia as well, and together with him, they managed to improve some of the results from the paper with Brech (like eliminating the hypothesis of continuum being regular). A joint work of A. Aviles and W. Kubis is currently on preparation.
Line 5 (collaboration with David Guerrero Sanchez)
David Guerrero Sanchez is a PhD student in Murcia under joint supervision of Antonio Aviles and Bernardo Cascales. Aviles and Guerrero Sanchez have been working in the problem whether a scattered subset of a Banach space in the weak topology must be sigma-discrete. This is a combinatorial problem related to the question of finding an intrinsic combinatorial characterisation of Banach spaces with a Kadets renorming. Some positive partial results are collected in a manuscript by Aviles and Guerrero Sanchez that is currently submitted for publication.
Line 6 (collaboration with A.J. Guirao and J. Rodriguez)
In the context of operator equations in Banach spaces, with J. Rodriguez from Murcia and Antonio Jose Guirao from Valencia, we are studying the so-called 'Bishop-Phelps-Bollobas property for numerical radius'. This is a property of a Banach space roughly stating that whenever an operator almost attains its numerical radius, it can be slightly perturbed so that it precisely attains its numerical radius. This is a work currently in preparation, and the main result that we have obtained so far is that every separable space of the form C(K) has this property.
(1) operator equations;
(2) homological methods in Banach spaces;
(3) renorming theory of Banach spaces;
(4) infinite-dimensional topology; and
(5) spaces of continuous functions on compact spaces.
The list of topics within the field where set theory, functional analysis and topology meet, and we could expect to make some progress, was expected to be expanded by the interaction with different experts.
The research activity can be organised along the following lines:
Line 1 (collaboration with G. Plebanek and J. Rodriguez)
After arriving in Murcia, the researcher initiated a collaboration with Jose Rodriguez from this university. He was working in a number of problems on functional analysis that seemed to have a set-theoretic flavor, and seemed thus to suit perfectly in the spirit of the project. One of our first decisions was to invite Grzegorz Plebanek to visit us and to discuss these problems, because, being an expert on abstract measure theory with a knowledge on Banach spaces and set-theory, he had a suitable profile that for the purposes of the problems we wanted to study. This was quite a success, and at the end we have four papers authored by the team Aviles-Plebanek-Rodriguez contributing to the success of this project. We briefly describe each of them.
1. 'The McShane integral in weakly compactly generated spaces' (J. Funct. Anal.)
In this paper, a problem posed by L. Di Piazza and D. Preiss is solved by constructing a function f:[0,1] --> X which is Pettis-integrable but not McShane integrable, yet X is a weakly compactly generated Banach space.
2. 'Measurability in C(2^k) and Kunen cardinals' (Israel J. Math.)
In this paper, a substantial improvent of a result of Fremlin is made, by showing that in the space C(K) of continuous functions on the generalised Cantor cube K = 2^k, the Baire and the Borel sigma-algebras coincide if and only if k is a Kunen cardinal.
3. 'A weak- separable C(K)- whose ball is not weak- separable' (Trans. A.M.S.)
In this paper, a space C(K) is constructed with the properties stated in the title and some measurability questions on it are also discussed. The only previously known example of such a space had been constructed by Talagrand assuming the continuum hypothesis, but this space exists in ZFC.
4. 'On Baire measurability in spaces of continuous functions' (J. Math. Anal. App.)
In this paper, under CH, a compact space K is constructed which has a probability measure which is in the w*-sequential closure of the convex hull of Dirac measures, but it is not the w*-limit a sequence of them.
Line 2 (collaboration with P. Koszmider)
One important open question of set-theoretic topological nature in nonseparable Banach space theory was whether the continuous image of a Radon-Nikodym compact space is again Radon-Nikodym compact. The researcher had worked on this topic in his PhD, and Piotr Koszmider from Warsaw got interested in the problem and in this previous work on it and he had some ideas to use the technique of resolutions of compact spaces to make some progress. A collaboration started, P. Koszmider visited Murcia on the fall of 2011 and a counterexample to the problem was finally obtained that has been recently accepted for publication in Duke Math. J. The result has important consequences in spaces of continuous functions, as it provides a space C(K) which is not isomorphic to C(L) with L zero-dimensional, despite the fact that K is 'very nice' (it has, for instance, lot of convergent sequences).
There is another joint work with P. Koszmider, in which, answering a question by Haily, Kaidi and Rodriguez-Palacios, a Banach space X is constructed, actually of the form C(K), such that every injective operator on X is an isomorphism.
Line 3 (collaboration with S. Todorcevic)
The researcher has been engaged with Stevo Todorcevic in developing the theory of multiple gaps, a topic on which they have written three papers during this period (one in Fund. Math, another in Combinatorica and the third is currently submitted for publication). The notion of multiple gap is a purely set-theoretic one, constituting a high dimensional generalisation of the classical concept of gap tracing back to Hausdorff. Aviles and Todorcevic invented this as a tool to deal with injectivity in Banach spaces of continuous functions, in the context of what we call 'homological methods in Banach spaces', in connection with some results of Ditor. However, it was realised later that there is a deep structural theory of these objects that has an interest on its own and quite different applications. This is particularly true in the case of analytic multiple gaps, to which the second and third paper are exclusively devoted. This analytic theory has found different applications in separable Banach space theory, as it allows to identify the canonical ways in which different basic sequences can be mixed in a single sequence. Part of this research was done during a visit of Antonio Aviles to Paris in the summer of 2011.
Line 4 (homological methods)
There have been actually some progress on the topic of 'homological methods in Banach spaces'. Christina Brech from Sao Paulo was invited to visit in Murcia, and this lead to a discussion about the results on the paper 'on Banach spaces of universal disposition' by Aviles et al. It was realised that, though the general language of category theory, one could connect problems in Banach space theory with problems on set theory considered in a paper of Dow and Hart. Putting all these ideas together, it was possible to answer some questions from both sides, and this gave origin to the paper 'A Boolean algebra and a Banach space obtained by push-out iteration' by Aviles and Brech. Later, Wieslaw Kubis from Prague, visited Murcia as well, and together with him, they managed to improve some of the results from the paper with Brech (like eliminating the hypothesis of continuum being regular). A joint work of A. Aviles and W. Kubis is currently on preparation.
Line 5 (collaboration with David Guerrero Sanchez)
David Guerrero Sanchez is a PhD student in Murcia under joint supervision of Antonio Aviles and Bernardo Cascales. Aviles and Guerrero Sanchez have been working in the problem whether a scattered subset of a Banach space in the weak topology must be sigma-discrete. This is a combinatorial problem related to the question of finding an intrinsic combinatorial characterisation of Banach spaces with a Kadets renorming. Some positive partial results are collected in a manuscript by Aviles and Guerrero Sanchez that is currently submitted for publication.
Line 6 (collaboration with A.J. Guirao and J. Rodriguez)
In the context of operator equations in Banach spaces, with J. Rodriguez from Murcia and Antonio Jose Guirao from Valencia, we are studying the so-called 'Bishop-Phelps-Bollobas property for numerical radius'. This is a property of a Banach space roughly stating that whenever an operator almost attains its numerical radius, it can be slightly perturbed so that it precisely attains its numerical radius. This is a work currently in preparation, and the main result that we have obtained so far is that every separable space of the form C(K) has this property.