Operator Algebra Seminar

Abstract: For countable discrete groups with property rapid decay(RD), Ebrahim Samei and other coauthors characterize amenable action on the stationary space using entropy. Recently under Das and Peterson’s noncommutative setting of Poisson boundaries and entropies, I generalized this result using similar arguments, and I will briefly introduce my recent work in this talk. Before introducing my recent work, I will first briefly introduce Das and Peterson’s noncommutative setting, which has been further developed by Zhou. Then I will show that for countable groups with property RD, amenability of W*-dynamical system implies the Furstenberg type entropy of the crossed product is maximal. In particular, if the action is state preserving, then these two conditions are equivalent to the condition that the group is amenable. After that for general countable groups, I will show that under some conditions, the reverse direction holds, in other words the maximality of the Furstenberg type entropy of the crossed product implies the W*-dynamical system is amenable.