Amenable actions and alegraic groups
Time and place
12:30–1:25 PM on Monday, June 3rd, 2024; Math Thesis Room 4214.03, Fourth Floor, CUNY Grad Center
Alain Valette (Univ. Neuchâtel)
Abstract
It was a crucial observation by R.J. Zimmer in the late '70's, that some measurable actions of non-amenable groups "look like" actions of amenable groups; those actions he termed "(measurably) amenable" (example: the action of $SL_2(\R)$ or $SL_2(\Z)$ on $P^1(\R)$ is amenable). The topological counterpart was proposed in 1987 by C. Anantharaman-Delaroche. Amenability of actions has a deep connection with nuclearity of the corresponding crossed product operator algebras (either von Neumann or C*-algebraic). The exact relationship between measurable and topological amenability was a longstanding open question that was solved only in 2022 by combining works by Buss-Echterhoff-Willett and Bearden-Crann. Using their solution, we observe that, for smooth actions on locally compact spaces (in the sense that the orbit space is countably separated), amenability of the action is equivalent to amenability of all stabilizers. By a result of Borel-Serre, a class of examples of such actions is algebraic actions of real or p-adic algebraic groups on algebraic varieties.
