Weekly Activities

A brief survey on partial actions
Speaker: Mikhailo Dokuchaev (Department of Mathematics, University of S\~ao Paulo, Brazil)

Time: 9h00, Wednesday, January 15, 2020,

Location: Room 611-612, Building A6, Institute of Mathematics

Abstract: Partial actions and partial representations were introduced in the theory of operator algebras as crucial ingredients of a new approach in the study of $C^{ast}$-algebras generated by partial isometries, permitting one to endow with the structure of a crossed product by a partial action such important classes of $C^{ast}$-algebras as the Cuntz-Krieger algebras, the Toeplitz algebras of quasi-lattice ordered groups, and more recently, the $C^*$-algebras of dynamical systems of type $(m,n),$ the ultragraph $C^*$-algebras and the Carlsen-Matsumoto $C^*$-algebras related to arbitrary subshifts. A remarkable recent application was achieved by P. Ara and R. Exel (2014), where for a finite bipartite separated graph the tame graph $C^*$-algebra was introduced and proved to be isomorphic to a partial crossed product of a commutative $C^*$-algebra by a finitely generated free group. This permitted the authors to give a negative answer to an open problem on paradoxical decompositions in a topological setting. In algebraic context partial actions were applied to graded algebras, Hecke algebras, Leavitt path algebras, Steinberg algebras, the Thompson group $V$, inverse semigroups, restriction semigroups and automata theory.