Exotic Ideals in Free Transformation Group $C^*$-Algebras

Abstract

Let $\Gamma$ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space $X$ and let $C(X) \rtimes_\eta \Gamma$ be the completion of the convolution algebra $C_c(\Gamma,C(X))$ with respect to a $C^*$-norm $\eta$. A non-zero ideal $J \unlhd C(X) \rtimes_\eta \Gamma$ is exotic if $J \cap C(X) = \{0\}$. We show that exotic ideals are present whenever $\Gamma$ is non-amenable and there is an invariant probability measure on $X$. 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 $\Gamma$ is a countably-infinite group having property (T) and $X$ is the Cantor set, there exists a free and minimal action of $\Gamma$ on $X$ and a $C^*$-norm $\eta$ on $C_c(\Gamma, C(X))$ such that $C(X)\rtimes_\eta\Gamma$ 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 $C(X)\rtimes_\eta \Gamma$ have trivial intersection with $C(X)$, and a result from arXiv:1901.09683 shows they coincide when the action of $\Gamma$ 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 $\Gamma$ on a compact Hausdorff space $X$ along with a $C^*$-norm $\eta$ for which these ideals are non-trivial, in particular, they are exotic ideals.