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FRINATEK-Fri prosj.st. mat.,naturv.,tek

C*-Algebraic Facets of Irreversible Dynamical Systems

Alternative title: C*-algebraiske fasetter av irreversible dynamiske systemer

Awarded: NOK 3.6 mill.

Surprisingly often, an elementary question triggers important development in search for sophisticated answers to it, for example: Why do we breathe? What happens inside a water stream? How can we model physical systems where the order of two successive measurements affects the outcome? The last question lead to the theory of C*-algebras arising from Heisenberg's matrix mechanics in the 1920s. It is a guiding theme for C*-algebraists to start from certain mathematical data and aim at constructing tractable objects which reflect key features of the initial data. For example, one can look at rotations of the circle. Thinking of the circle as the interval [0,1] with the endpoints glued together, rotation by some angle is nothing but adding a certain real number r modulo 1. This setting really is interesting for irrational r. In 1981, Rieffel basically showed that r can be recovered from its C*-algebra. While this is an example with a single, reversible transformation, this project focuses on irreversible transformations. Moreover, we are interested in dynamical systems whose complexity increases significantly when multiple transformations are involved. A first example for this is given by replacing rotation on the circle by squaring and cubing. Basically, this is the same as multiplying integers by two and three. Already for such elementary systems, surprising phenomena occur as observed by Furstenberg in 1967. His results on topological properties related to orbit closures made him conjecture a rigidity of invariant probability measures. This conjecture is still open despite the efforts of many experts in the field. In order to understand in how far C*-algebraic models may help to resolve the Furstenberg conjecture, this project aims at a careful examination of several C*-algebras that are quite natural to consider for such a dynamical system. But the analysis is carried out for more general dynamical systems and the results are of interest for different disciplines. This project has made significant contributions to the establishments of a new research line "C*-algebras of right LCM monoids". Starting off from dynamical systems involving injective group endomorphisms, the framework of right LCM monoids was subsequently identified as the proper (& more general) setting, in which striking results were obtained on - the consistency of the various ways to associate C*-algebras to the initial data; - the detection of internal structures of right LCM monoids that are relevant to the features of their C*-algebras; - the determination of classifying invariants (topological K-theory), leading to an intricate conjecture relating to single-vertex higher-rank graphs; and - the classification of equilibrium states (so-called Kubo-Martin-Schwinger states) for dynamical systems, where, in the absence of a state space, the respective C*-algebra represents the ensemble of observables. As a side effect of the project, we are sensing a rising mutual interest for exchange & collaborations between semigroup theorists and C*-algebraists. Seen in the context of recent progress in the study of operator algebras associated to left cancellative small categories, a great share of the results gives indications as to what may be true in this very broad setting. The project manager is grateful to the Research Council of Norway for its support for the project. He hopes that the RCN is pleased with its performance, and will revert to its long tradition of a balanced funding spectrum across the hypothetical borders of applied and theoretical research for the sake of sustainability and long-range impacts which may not be measurable in proximal industrial/societal appliances and products.

Outcomes: 1) The workgroup has become a vibrant hub for research on operator algebras in connection with semigroups and groupoids. 2) Professor Nadia S. Larsen and the project manager became key figures behind the establishment of the research line "operator algebras of right LCM monoids" with many opportunities for further research. 3) The project manager gained valuable experience with regards to international experience, project management, organization of scientific events, forging new alliances & collaborations, and prioritization of tasks and targets. Impact: 1) The workgroup's global scientific network has been strengthened and expanded, in particular with respect to the nodes in Australia(Sydney,Wollongong,Newcastle), Canada(Victoria), and UK(Edinburgh,Glasgow). 2) The project's results suggest that a stronger interdisciplinary exchange between operator algebraists, semigroup theorists, and groupoidists would bear the potential for great advances.

Surprisingly often, an elementary question triggers important development in search for sophisticated answers to it, for example: Why do we breathe? What happens inside a water stream? How can we model physical systems where the order of two successive measurements affects the outcome? The last question lead to the theory of C*-algebras arising from Heisenberg's matrix mechanics in the 1920s. It is a guiding theme for C*-algebraists to start from certain mathematical data and aim at constructing tractable objects which reflect key features of the initial data. For example, one can look at rotations of the circle. Thinking of the circle as the interval [0,1] with the endpoints glued together, rotation by some angle is nothing but adding a certain real number r modulo 1. This setting really is interesting for irrational r. In 1981, Rieffel basically showed that r can be recovered from its C*-algebra. While this is an example with a single, reversible transformation, this project focuses on irreversible transformations. Moreover, we are interested in dynamical systems whose complexity increases significantly when multiple transformations are involved. A first example for this is given by replacing rotation on the circle by squaring and cubing. Basically, this is the same as multiplying integers by two and three. Already for such elementary systems, surprising phenomena occur as observed by Furstenberg in 1967. His results on topological properties related to orbit closures made him conjecture a rigidity of invariant probability measures. This conjecture is still open despite the efforts of many experts in the field. In order to understand in how far C*-algebraic models may help to resolve the Furstenberg conjecture, this project aims at a careful examination of several C*-algebras that are quite natural to consider for such a dynamical system. But the analysis is carried out for more general dynamical systems and the results are of interest for different disciplines.

Funding scheme:

FRINATEK-Fri prosj.st. mat.,naturv.,tek