2021/09/13 by Ruy Exel, David R. Pitts, Exel, Ruy +3
Mathematics · #46L55 #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Mathematical and Theoretical Analysis #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.2109.06293
openalex publication_date 2021/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a discrete group acting freely via homeomorphisms on the compact Hausdorff space X and let C(X) \rtimesηΓ be the completion of the convolution algebra Cc(Γ,C(X)) with respect to a C^*-norm η. A non-zero ideal J \unlhd C(X) \rtimesηΓ is exotic if J ∩ C(X) = \0\. We show that exotic ideals are present whenever Γ 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 Γ is a countably-infinite group having property (T) and X is the Cantor set, there exists a free and minimal action of Γ on X and a C^*-norm η on Cc(Γ, C(X)) such that C(X)\rtimesηΓ 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)\rtimesηΓ have trivial intersection with C(X), and a result from arXiv:1901.09683 shows they coincide when the action of Γ 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 Γ on a compact Hausdorff space X along with a C^*-norm η for which these ideals are non-trivial, in particular, they are exotic ideals.