International collaboration to advance research in operator algebras

Faculty of Science
Mathematics and statistics
Professor Aaron Tikuisis, wearing checkered mint green and white shirt, is standing in front of the living wall in the FSS building on uOttawa campus
Canada holds an important place on the international operator algebras stage. The subject is well-represented by a vibrant community of researchers at several Canadian universities, including many international leaders.

Professor Aaron Tikuisis is a well-respected researcher in the area of operator algebras. He recently co-authored three peer-reviewed research papers in partnership with colleagues Jorge Castillejos (Associate Researcher, Universidad Nacional Autónoma de México), Samuel Evington (postdoctoral researcher, University of Münster, Germany), Stuart White (Professor of Mathematics, University of Oxford, UK) and Wilhelm Winter (Professor of Mathematics, University of Münster, Germany).

Operator algebras is a mathematical subject that arose to provide a rigorous framework for quantum mechanics. It comprises two branches: von Neumann algebras and C*-algebras, and throughout the subject, non-commutativity forms a central theme – that is, that the order in which measurements are taken matters to the outcome. The Elliott classification program, a core part of C*-algebras research for the past 30+ years, aims to systematically classify suitably small C*-algebras, using computable invariants. Prof. Tikuisis’ collaborative work was motivated by the Toms-Winter conjecture, stemming from the Elliott classification program, which postulates the equivalence of three conceptually different properties (one function-analytic, one topological, and one cohomological) of C*-algebras.

Along with his team of international collaborators, Prof. Tikuisis proved one implication of the Toms-Winter conjecture, by developing a new technique for transferring results from von Neumann algebras to C*-algebras. Von Neumann algebras forms the other main branch (alongside C*-algebras) of operator algebra theory, and Prof. Tikuisis’ team included experts in both von Neumann algebras and C*-algebras. Follow-up work, including ongoing projects, involve finding other applications of this new technique.

The Toms-Winter conjecture has been a key part of Prof. Tikuisis’ research program over the last several years. The conjecture, and its relation to the Elliott classification program, has created considerable interest in the operator algebras community. Indeed, Prof. Tikuisis and his team devoted significant time working on the problem, through various attempts, before they finally developed the right approach. Excitement around the result has grown, particularly as it is driving a new approach to the classification of C*-algebras, one that brings in von Neumann algebra theory.

Such collaborative efforts with international scientists contribute to bringing visibility to Canadian scientists, and to the University of Ottawa, on the world stage. The Canadian operator algebras community and its international esteem have been especially strengthened by the annual Canadian Operator Symposium, whose 50th anniversary edition was held at uOttawa in 2022, co-organized by Prof. Tikuisis.

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