Exotic ideals in free transformation group C∗$\mathbf {C}^*$‐algebras

Abstract

AbstractLet be a discrete group acting freely via homeomorphisms on the compact Hausdorff space and let be the completion of the convolution algebra with respect to a ‐norm . A non‐zero ideal is exotic if . We show that exotic ideals are present whenever is non‐amenable and there is an invariant probability measure on . This fact, along with the recent theory of exotic crossed product functors, allows us to provide answers to two questions of K. Thomsen.Using the Koopman representation and a recent theorem of Elek, we show that when is a countably‐infinite group having property (T) and is the Cantor set, there exists a free and minimal action of on and a ‐norm on such that contains the compact operators as an exotic ideal. We use this example to provide a positive answer to a question of A. Katavolos and V. Paulsen.The opaque and grey ideals in have trivial intersection with , and a result from Exel and Pitts [Characterizing groupoid ‐algebras of non‐Hausdorff étale groupoids, Lecture Notes in Mathematics, vol. 2306, Springer, Cham, 2022] shows that they coincide when the action of is free; however, the problem of whether these ideals can be non‐zero was left unresolved. We present an example of a free action of on a compact Hausdorff space along with a ‐norm for which these ideals are non‐trivial, in particular, they are exotic ideals.