Final Report Summary - DSTOA (Descriptive set theory and operator algebras)
Background: DSTOA is a research project in mathematics funded by the European Union (EU) through the Seventh Framework Programme and carried out at the University of Vienna (until 30 June 2011) and at the University of Copenhagen (from 1 July 2011) by Asger Tornquist (researcher), Sy-David Friedman and Mikael Rordam (scientists in charge). The acronym DSTOA stands for 'descriptive set theory and operator algebras'. Descriptive set theory is an area of mathematics which is concerned with how complex different mathematical objects or problems are and is a theoretical subject that belongs to the foundations of mathematics. Operator algebras, on the other hand, is a large area within mathematics which grew out of the need to understand linear operators on spaces (generally of infinite dimension), a need that arose in the 1930s in quantum physics, among other places. The two areas involved in the research project have co-existed with only sporadic interaction until about 10 years ago, when interests within the two fields converged. This has turned out to be an unusually fruitful turn of events, as techniques in one area has turned out to be useful for solving problems in the other and vice versa. It should be mentioned that this convergence of interests in mathematics is broader than what is indicated here: In today's mathematics there is a broad interests in group actions, orbit equivalence relations and rigidity phenomena which is shared among researchers not just in operator algebras and descriptive set theory, but naturally also by researchers in ergodic theory, dynamical systems, measurable group theory and geometric group theory. The research in these other areas is also of great importance to the goals of this project.
The project DSTOA takes aim at several important problems within this field of research; in particular it has the following research objectives:
1. what is the Borel reducibility complexity of orbit equivalence and von Neumann equivalence?
2. what is the hierarchy complexity of these and other key equivalence relations in operator algebras?
3. what effect does extra set-theoretic axioms have on the structure and liftings of quotient objects in operator algebras and ergodic theory?
Results:
The earliest results of the project all addressed questions related to objective one above. A specific goal of the project was to show that ITPFI factors do not admit classification by countable structures and this was goal was achieved in collaboration with Roman Sasyk (Buenos Aires). Another major result in this direction was achieved with Inessa Epstein from Caltech, Pasadena, United States of America (USA), where it was shown that orbit equivalence and von Neumann equivalence are for most non-amenable groups not Borel, but true analytic equivalence relations. This result is one of the technically most difficult achievements of the project.
Objective two turned out to be the most surprising and successful part of the project. In the original proposal, the idea of addressing the classification problem of separable C*-algebras through set theoretic considerations was presented as a 'hope for the distant future'. Well, the future has arrived quickly! In collaboration with an array of people, but most importantly Andrew Toms and Ilijas Farah, we were able to achieve a string of remarkable results and make vast progress. We showed that the key class of C*-algebras, the nuclear separable simple ones, does not admit classification by countable structures and in fact, the classification problem is extremely complex, as measured by Borel reducibility. We showed that the computation of invariants of C*-algebras are Borel; and most recently, we have shown, in collaboration with also Elliott, Paulsen and Rosendal, that isomorphism relation of general separable C*-algebras is below a group action, giving a surprising upper bound on complexity.
The third objective was completely settled, if one is willing to assume the so-called continuum hypothesis (CH), a statement about the size of subsets of the real numbers that is consistent with mathematics today, but is not generally accepted as part of the foundations of mathematics. Specifically, we showed that assuming CH, every near-action can be implemented by Borel automorphisms. It was then the hope that one could show that in a model of set theory where CH fails, this result may also fail. In this direction, only a partial result was achieved (together with J. Kellner, KGRC, University of Vienna). We showed that if there are certain very special pointwise implementations of near-actions, then the measure-algebra has a lifting. This then gives that in a model constructed by Shelah where the measure algebra has no lifting, there are near-actions that don't admit 'special' liftings. We are optimistic that these ideas will, with further work, result in a full resolution of this sub-objective of objective three.
The project DSTOA takes aim at several important problems within this field of research; in particular it has the following research objectives:
1. what is the Borel reducibility complexity of orbit equivalence and von Neumann equivalence?
2. what is the hierarchy complexity of these and other key equivalence relations in operator algebras?
3. what effect does extra set-theoretic axioms have on the structure and liftings of quotient objects in operator algebras and ergodic theory?
Results:
The earliest results of the project all addressed questions related to objective one above. A specific goal of the project was to show that ITPFI factors do not admit classification by countable structures and this was goal was achieved in collaboration with Roman Sasyk (Buenos Aires). Another major result in this direction was achieved with Inessa Epstein from Caltech, Pasadena, United States of America (USA), where it was shown that orbit equivalence and von Neumann equivalence are for most non-amenable groups not Borel, but true analytic equivalence relations. This result is one of the technically most difficult achievements of the project.
Objective two turned out to be the most surprising and successful part of the project. In the original proposal, the idea of addressing the classification problem of separable C*-algebras through set theoretic considerations was presented as a 'hope for the distant future'. Well, the future has arrived quickly! In collaboration with an array of people, but most importantly Andrew Toms and Ilijas Farah, we were able to achieve a string of remarkable results and make vast progress. We showed that the key class of C*-algebras, the nuclear separable simple ones, does not admit classification by countable structures and in fact, the classification problem is extremely complex, as measured by Borel reducibility. We showed that the computation of invariants of C*-algebras are Borel; and most recently, we have shown, in collaboration with also Elliott, Paulsen and Rosendal, that isomorphism relation of general separable C*-algebras is below a group action, giving a surprising upper bound on complexity.
The third objective was completely settled, if one is willing to assume the so-called continuum hypothesis (CH), a statement about the size of subsets of the real numbers that is consistent with mathematics today, but is not generally accepted as part of the foundations of mathematics. Specifically, we showed that assuming CH, every near-action can be implemented by Borel automorphisms. It was then the hope that one could show that in a model of set theory where CH fails, this result may also fail. In this direction, only a partial result was achieved (together with J. Kellner, KGRC, University of Vienna). We showed that if there are certain very special pointwise implementations of near-actions, then the measure-algebra has a lifting. This then gives that in a model constructed by Shelah where the measure algebra has no lifting, there are near-actions that don't admit 'special' liftings. We are optimistic that these ideas will, with further work, result in a full resolution of this sub-objective of objective three.