Periodic Reporting for period 1 - IPOA (Independence Phenomena in Operator Algebras)
Reporting period: 2020-11-01 to 2022-10-31
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.
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.
One of the most influential applications of mathematical logic to operator algebras has been the study the existence of outer automorphisms of the Calkin algebra Q(H). While the motivation behind this work are purely of C*-algebraic nature, the solution turned out to be heavily influenced by set theory. More precisely, the existence of outer automorphisms of Q(H) is independent from the Zermelo-Fraenkel axiomatization of set theory: in 2007 Phillips and Weaver proved that under the Continuum Hypothesis it is possible to build outer automorphisms of Q(H), while Farah in 2011 showed that the Open Coloring Axiom entails that all automorphisms of Q(H) are inner. During the early months of the project, the researcher expanded the scope of this analysis to the set of endomorphisms of the Calkin algebra, obtaining a clean generalization of the results obtained by Farah. This work led to a complete classification of the set of unital endomorphisms of Q(H) up to unitary equivalence, as well as some interesting consequences on the structure of the class of C*-algebras that embed into the Calkin algebra.
The aforementioned work is part of a broader line of research investigating how set-theoretic axioms influence the nature and the rigidity of massive quotient structures in various categories, such as Boolean algebras, topological spaces and operator algebras. The researcher participated in the writing of an expository survey on this topic, a joint work with Ilijas Farah, Saeed Ghasemi and Alessandro Vignati. This survey aims to be an exhaustive point of reference for researchers who approach this subject.
The main collaboration produced by the researcher in the course of this project with other researcher in Paris consists of an application of infinitary model theory to the study of C*-algebras which fall within the scope of Elliott Classification Program. The latter refers to one of the most active and vibrant areas of research in operator algebras, aiming to classify sufficiently regular C*-algebras (usually referred to as classifiable C*-algebras) by computable invariants arising from the K-theory of a C*-algebra. One of the key methods in this program are the intertwining arguments allowing to build isomorphisms between C*-algebras starting from isomorphisms between the invariants. With other collaborator in Paris, the researcher conducted a deep analysis of how these arguments relate and combine with back-and-forth methods and Ehrenfeucht-Fraïssé games coming from model theory, focusing on approximately finite C*-algebras (AF-algebras). This connection proved to be extremely fruitful, leading to an extremely clear picture of how the model theoretic features of an AF-algebra are determined by those of its corresponding Elliott Invariant, which in this case reduces to its dimension group.
In the course of this project, the researcher included in his work topics which are currently central in the community of operator algebraists, focusing in particular on problems disjoint from logic and arising in the context of C*-dynamics and in the study of crossed products of C*-algebras. This effort led to a series of paper and significant advancements in the state of the art, with various researchers from University of Muenster, Gothenburg University and Ben-Gurion University of the Negev.
The aforementioned work is part of a broader line of research investigating how set-theoretic axioms influence the nature and the rigidity of massive quotient structures in various categories, such as Boolean algebras, topological spaces and operator algebras. The researcher participated in the writing of an expository survey on this topic, a joint work with Ilijas Farah, Saeed Ghasemi and Alessandro Vignati. This survey aims to be an exhaustive point of reference for researchers who approach this subject.
The main collaboration produced by the researcher in the course of this project with other researcher in Paris consists of an application of infinitary model theory to the study of C*-algebras which fall within the scope of Elliott Classification Program. The latter refers to one of the most active and vibrant areas of research in operator algebras, aiming to classify sufficiently regular C*-algebras (usually referred to as classifiable C*-algebras) by computable invariants arising from the K-theory of a C*-algebra. One of the key methods in this program are the intertwining arguments allowing to build isomorphisms between C*-algebras starting from isomorphisms between the invariants. With other collaborator in Paris, the researcher conducted a deep analysis of how these arguments relate and combine with back-and-forth methods and Ehrenfeucht-Fraïssé games coming from model theory, focusing on approximately finite C*-algebras (AF-algebras). This connection proved to be extremely fruitful, leading to an extremely clear picture of how the model theoretic features of an AF-algebra are determined by those of its corresponding Elliott Invariant, which in this case reduces to its dimension group.
In the course of this project, the researcher included in his work topics which are currently central in the community of operator algebraists, focusing in particular on problems disjoint from logic and arising in the context of C*-dynamics and in the study of crossed products of C*-algebras. This effort led to a series of paper and significant advancements in the state of the art, with various researchers from University of Muenster, Gothenburg University and Ben-Gurion University of the Negev.
The introduction of methods from infinitary logic considered in this project led to a completely novel approach when it comes to the study of C*-algebras via continuous model theory. The results obtained in the paper 'Games on AF-algebras' largely expand the previous knowledge on the relations between a C*-algebra and its K0-group (from the perspective of model theory), which before this effort were mainly compose by scattered and partial statements. The ideas introduced in the paper 'Games on AF-algebra' provide a versatile template that could be conceivably replicated on larger classes of classifiable C*-algebras. In this more general setting, the dimension group is no longer sufficient to classify C*-algebras up isomorphism, and it is substituted by a more complex and refined object, the so-called Elliott Invariant. This endeavor could also facilitate a more systematic study of infinitary logic in the continuous setting, as developed in the paper 'Metric Scott analysis' by Yaacov, Doucha, Nies and Tsankov.
The results produced over the two years of the present project, for what concerns C*-dynamics and crossed products of classifiable C*-algebras, largely expanded the previous knowledge on the topic and represent the current state of the art. More precisely, the theorems presented in the papers 'Strongly outer actions of amenable groups on Z-stable nuclear C*-algebras' and 'Dynamical comparison and Z-stability for crossed products of simple C*-algebras' led to significant advancements for what concerns our understanding of equivariantly Z-stable actions. The research developed over the course of the present project, and its continuation envisioned for the upcoming years, can be expected to be relevant also for the classification, up to cocycle conjugacy, of actions of amenable groups on classifiable C*-algebras by suitable computable invariants, in analogy to what was done for actions on von Neumann algebras by Connes, Jones and Ocneanu.
The results produced over the two years of the present project, for what concerns C*-dynamics and crossed products of classifiable C*-algebras, largely expanded the previous knowledge on the topic and represent the current state of the art. More precisely, the theorems presented in the papers 'Strongly outer actions of amenable groups on Z-stable nuclear C*-algebras' and 'Dynamical comparison and Z-stability for crossed products of simple C*-algebras' led to significant advancements for what concerns our understanding of equivariantly Z-stable actions. The research developed over the course of the present project, and its continuation envisioned for the upcoming years, can be expected to be relevant also for the classification, up to cocycle conjugacy, of actions of amenable groups on classifiable C*-algebras by suitable computable invariants, in analogy to what was done for actions on von Neumann algebras by Connes, Jones and Ocneanu.