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9 czerwca 2026 / June 9, 2026
Piotr M. Hajac (IMPAN - Warsaw - Poland)
The art of moves on graphs and extended covariant functoriality of graph algebras
Abstract: Combinatorics of graphs is a very powerful tool to unravel various properties of graph algebras. In particular, isomorphisms between graph algebras are often implemented by moves between their graphs. In this talk, I will explain how to make these combinatorial methods functorial. In particular, I will show that collapsing an outsplitted graph to the original graph, or transforming a graph to a shifted graph, can be implemented by admissible graph or path homomorphisms, respectively. To include the inverses of such isomorphisms, I will introduce a new category of graphs where morphisms are given as regular homomorphisms of graph inverse semigroups. This new category admits a covariant functor to the category of C*-algebras and *-homomorphisms which extends the known covariant functor from the category of graphs and admissible path homomorphisms. I will demonstrate the new framework by applying it to various graphs that yield the minimal unitization of compact operators (the standard Podleś quantum sphere) and matrices over Cuntz algebras. (Based on joint work with G. G. de Castro, M. Lowiel, and E. Pacheco.)