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11 czerwca 2024 / June 11, 2024
Piotr M. Hajac (IMPAN)
Noncommutative topology through operator algebras that one can see
Abstract. This is an introductory talk that will start with unravelling the paradigm of noncommutative topology followed by a step-by-step construction of graph C*-algebras abundantly instantiated. Due to their tangible combinatorial nature, these C*-algebras are known as "operator algebras that one can see". The main part of the talk will be devoted to a conceptual explanation (technicalities omitted, motivating examples provided) of how we solved the mixed-functoriality problem of pullbacks of graph C*-algebras. More precisely, while some pullbacks can be understood as coming from pushouts of directed graphs via a contravariant functor assigning C*-algebras to graphs, other pullbacks require both covariant and contravariant functors assigning C*-algebras to graphs considered as objects in two different categories. In the latter case, one cannot explain the pullback structure of a graph C*-algebra in terms of a commuting diagram of underlying graphs in one category of graphs. We solve this problem by introducing a new category of directed graphs, where morphisms are relations rather than maps, and define a covariant functor from this category to the category of C*-algebras. Now, the new functor generalizes both the covariant and contravariant functors used before. Our main result is a general pullback theorem including both types of the aforementioned pullbacks as special cases. Time permitting, the talk will end by posing a tantalizing problem involving both pullbacks of noncommutative C*-algebras and the Poincaré conjecture. (Based on recent joint works with Mariusz Tobolski and Gilles Gonçalves de Castro.)