Exotic Ideals in Represented Free Transformation Groups

Let $Gamma$ be a discrete group freely acting 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 $J unlhd C(X)rtimes_etaGamma$ is exotic if $Jcap C(X) =(0)$. We give examples showing 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 negative answers to two questions of K. Thomsen. Let $mathfrak{A}$ be a non-atomic MASA on a separable Hilbert space and let $mathcal B_0$ be the linear span of the unitary operators $Uinmathcal B(mathcal H)$ such that $Umathfrak{A} U^*=mathfrak{A}$. We observe that while $mathcal B_0$ contains no compact operators, the norm-closure of $mathcal B_0$ contains all compact operators. This gives a positive answer to a question of A. Katavolos and V. Paulsen. For a free action of $Gamma$ on a compact Hausdorff space $X$, the opaque and grey ideals in $C(X)rtimes_eta Gamma$ coincide. We conclude with an example of a free action of $Gamma$ on a compact Hausdorff space $X$ along with a norm $eta$ for which these ideals are non-trivial.