IPOA · Independence Phenomena in Operator Algebras
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-11-01 → 2022-10-31
- EU contribution
- €184,708
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Independence Phenomena in Operator Algebras
The general goal of this project is expanding the current knowledge on applications of mathematical logic, mainly in the form of set theory and model theory, to algebras of operators on Hilbert spaces, such as C*-algebras. The interaction between these disciplines has flourished over the last 20 years, after the groundbreaking work by Phillips-Weaver and Farah on the existence of outer automorphisms of the Calkin algebra. The present project aims both to expand and continue the existing lines of research on this topic, but also to explore how logic could bring novel ideas and approaches in the study of C*-algebras. The overall objectives are concretely represented by a series of goals concerning various topics in operator algebras where applications of set-theoretic and model-theoretic methods have already proven to be successful and fruitful, as well as new problems and questions. These include - the study of massive quotient structures, such as corona algebras and Cech-Stone reminders; - the employment of combinatorial set-theoretic statements and game-like techniques for the construction of interesting C*-algebras; - the study of automorphism groups and of dynamical systems of C*-algebras.
Data: CORDIS, © European Union
Project objective
This proposal develops in the framework of applications of set theory to C*-algebras and it is organized into three main themes: (1) the set-theoretic study of the Calkin algebra, (2) Naimark's problem, (3) the Stone-Weierstrass problem for noncommutative C*-algebras. The first part of the project consists of a systematic analysis of the class of the C*-algebras which embed into the Calkin algebra and of how set-theoretic principles influence such class. This study will be achieved by means of forcing techniques and through the adaptation of methods coming from the framework of boolean algebras. The main objectives are to reach a deeper understanding of the structure of the Calkin algebra, and to provide a benchmark for future applications of forcing methods in a more abstract C*-algebraic context. The second part of the proposal is in continuity with the line of research opened by Akemann and Weaver in the study of Naimark's problem, and it involves a series of applications of set-theoretic combinatorial statements in the construction of nonseparable C*-algebras with peculiar properties, specifically for what concerns their representation theory. With these investigations we aim to extend, by means of set theory, the current knowledge on the discrepancies between the nonseparable and the separable framework in operator algebras. The last part of the project regards the Stone-Weierstrass problem for noncommutative C*-algebras, an old open question which asks whether the classical Stone-Weierstrass theorem can be generalized to all C*-algebras. We plan to study this topic using set-theoretic methods, with the objective to find new consistency results, and extend to the nonseparable setting the known theorems holding for separable C*-algebras.
Original text from CORDIS.
Participants
- UNIVERSITE PARIS CITE · ParisCoordinatorFrance
Links
Data: CORDIS, © European Union