GRAPH MORPHISMS AS GROUPOID ACTORS

Speaker(s)

RALF MEYER

Affiliation

Universität Göttingen, Germany

Language of the talk

English

Date

May 6, 2026, 5:15 p.m.

Information about the event

IMPAN - Room 405 & zoom

Title in Polish

GRAPH MORPHISMS AS GROUPOID ACTORS

Seminar

North Atlantic Noncommutative Geometry Seminar

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A graph C*-algebra is known to be the groupoid C*-algebra of a certain groupoid, called the boundary-path groupoid. The boundary-path groupoid of a graph C*-algebra also has a universal property that is analogous to the universal property of the graph C*-algebra. This property specifies its category of actions on spaces. A groupoid actor can be defined as a functor between the categories of groupoid actions for two groupoids that act on the same underlying space. Groupoid actors can be described concretely through the action of one groupoid on the arrow space of the other groupoid. They induce nondegenerate morphisms between the groupoid C*-algebras. With an extra properness condition, we even get nondegenerate *-homomorphisms between the groupoid C*-algebras. In a recent joint article with de Castro, we described the proper groupoid actors from the boundary-path groupoid of a graph to any etale groupoid H. The result is a dynamical analogue of the familiar Cuntz-Krieger families in a C*-algebra, replacing partial isometries by compact open bisections in H. These dynamical Cuntz-Krieger families induce *-homomorphisms between the graph C*-algebras. When the target H is the boundary-path groupoid of another graph, this generalises the relation morphisms considered in earlier work by de Castro, D'Andrea and Hajac. Groupoid actors have the advantage that they are the arrows in a category, that is, they may be composed. In recent work with Raum and Taylor, we also show that a proper groupoid actor is invertible as an actor once it induces an isomorphism between the groupoid C*-algebras. Hence, the inverse of a *-homomorphism between graph C*-algebras that is induced by a dynamical Cuntz-Krieger family is also of this form.

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