Unitary Conjugation Groupoid Conjecture

Let $A$ be a separable unital $C^*$-algebra, and let $\mathcal{G}_A$ be its unitary conjugation groupoid, with unit space

$$\mathcal{G}_A^{(0)}\cong\widehat{A},$$

where $\widehat{A}$ consists of irreducible representations up to unitary equivalence. Assume the associated groupoid $C^*$-algebra has the Morita equivalence

$$C^*(\mathcal{G}_A)\sim_M A\otimes\mathcal{K}.$$

Unitary Conjugation Groupoid Conjecture. There exists a natural assembly map

$$\mu_{\mathcal{G}_A}:K^*_{\mathcal{G}_A}(\mathcal{G}_A^{(0)})\longrightarrow K_*(C^*(\mathcal{G}_A))$$

which is an isomorphism for a large class of $C^*$-algebras, including all nuclear $C^*$-algebras with Cartan subalgebras and all crossed products $C(X)\rtimes\Gamma$ with $\Gamma$ satisfying the Baum–Connes conjecture.

Under the Morita identification $K_*(C^*(\mathcal{G}_A))\cong K_*(A)$, this predicts that the analytic $K$-theory of $A$ can be recovered from equivariant topological data associated with the unitary conjugation groupoid. The source presents this as a conjectural framework and gives no resolution status.