Existence of infinite-dimensional crossed-product algebras for the Higson–Kasparov–Trout algebra

Let $U$ be the infinite-dimensional group under consideration, and let $\mathscr{SC}(U)$ be its associated Higson–Kasparov–Trout algebra. The notation $\underline{\mathscr{SC}(U)\rtimes U}$ and $\underline{\mathscr{SC}(U)\rtimes_\tau U}$ denotes two proposed $C^*$-algebras associated with the ordinary and twisted crossed products, respectively.

Existence conjecture. Two $C^*$-algebras playing the roles of $\mathscr{SC}(U)\rtimes U$ and $\mathscr{SC}(U)\rtimes_\tau U$ can be defined, namely $\underline{\mathscr{SC}(U)\rtimes U}$ and $\underline{\mathscr{SC}(U)\rtimes_\tau U}$.

The paper notes that there are no rigorous results in this subsection. Candidate constructions include $\mathscr{S}\otimes\mathcal{K}(\underline{L^2(U)}\otimes S^*)$ and $\mathcal{K}(\underline{L^2(U)})\otimes{\rm CAR}$, but the appropriate choice remains unresolved.