Jean Louis Tu showed that the (maximal) groupoid $C^*$-algebra of a groupoid satisfying the Haagerup property (which includes all amenable groupoids) will satisfy the UCT. I am curious if there are known examples of nuclear c*-algebras which satisfy the UCT and are definitely not groupoid $c^*$-algebras of an amenable etale groupoid. More generally, are there examples of c*-algebras which satisfy the UCT that are not groupoid $c^*$-algebras of Haagerup groupoids?
2026-02-22 19:36:40.1771789000
Are there examples of unital and nuclear $C^*$-algebras satisfying the UCT that are not groupoid algebras of an amenable etale groupoid?
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I just came across https://arxiv.org/pdf/1802.01190.pdf by Xin Li. For simple separable unital and finite nuclear dimension c*-algebras, he showed that satisfying the UCT is equivalent to being a twisted groupoid c*-algebra of an amenable etale groupoid. That more or less settles that.