# 北海道力学系webセミナー: Operator algebras and measure-theoretic properties on self-similar groups (吉田啓佑（北大理）)

2021年514日 開催

Time：2021年5月14日(金)　13:00～14:00

Speaker：吉田啓介（北大理）

Title： Operator algebras and measure-theoretic properties on self-similar groups

Abstract：Self-similar groups are some kinds of groups consisting of homeomorphisms on the Cantor space (we fix a finite set $X$ and identify the Cantor space with $X^\mathbb{N}$). By definition, self-similar groups provide some relations between elements in the group and one sided shift maps. Nekrashevych constructed C$^*$-algebras respecting the relations. We consider the Bernoulli measure on the Cantor space and observe measure-theoretic properties strongly affecting the C$^*$-algebras. More precisely, we show that two properties on C$^*$-algebras are closely related to the sets of fixed points of elements in self-similar groups. The first one is the uniqueness of the tracial states (on gauge invariant C$^*$-subalgebras). The second one is the simplicity (i.e. no non-trivial closed ideal). When discussing the second property, we only consider self-similar groups referred to as multispinal groups.

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