Groups and Operator Algebras Seminar

Speaker: Dan Ursu

Title: Intermediate subalgebras for reduced crossed products of discrete groups

Abstract: In joint work with Matthew Kennedy, we consider the problem of characterizing when the subalgebras of a reduced crossed product $A \rtimes_r G$ are canonical, restricting our attention to intermediate subalgebras of the form $A \subseteq B \subseteq A \rtimes_r G$. The "canonical" subalgebras to consider arise from partial subactions of $G \curvearrowright A$, and can be thought of as generalizations of subalgebras of the form $A \rtimes_r H$ for $H \leq G$.

We obtain a nearly complete, two-way characterization on when all such subalgebras are of this form, modulo some mild assumptions. In the case of commutative $A = C(X)$, this condition ends up being freeness of the action of $G$ on $X$. For the noncommutative setting, we needed to identify the "correct" notion of freeness of an action of $G$ on $A$, of which several already exist in the literature. The techniques involved are also quite different from those used in earlier results in the literature, and are more akin to those involved in studying simplicity of $A \rtimes_r G$. In particular, we make heavy use of the dynamics on injective envelope $I(A)$, a sort of "noncommutative boundary" of $A$, and also rely on some of the noncommutative convexity results of Magajna.