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Existence of C*-algebras failing the K-homology UCT

Identify whether there exist separable C*-algebras that do not satisfy Brown’s K-homology Universal Coefficient Theorem, and if so, construct explicit examples; otherwise, prove that all separable C*-algebras satisfy the K-homology UCT.

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Background

The paper relies on an approximate K-homology UCT (a weakened, topological version of the KK-UCT) and discusses relations among the KK-UCT, the K-homology UCT, and the approximate K-homology UCT. While the KK-UCT is known to fail in some cases, the status of the K-homology UCT is unclear.

Resolving whether any separable C*-algebra fails the K-homology UCT would clarify the strength and scope of UCT assumptions used in the paper’s stable uniqueness framework.

References

I do not know if there are $C*$-algebras that do not satisfy the $K$-homology UCT.

— Conditional representation stability, classification of $*$-homomorphisms, and relative eta invariants (2408.13350 - Willett, 23 Aug 2024) in Remark \ref{auct rem}, Section “Controlled K-homology, KL-theory, and total K-theory”