Conjecture on the cyclic cohomology of the smooth dihedral group algebra

Let $A^{\infty}$ be the smooth subalgebra of the group $C^*$-algebra associated with the group considered in the paper. For a cyclic cohomology degree $n$, write $HC^n(A^{\infty})$ for the corresponding cyclic cohomology group.

Cyclic cohomology conjecture. The cyclic cohomology of $A^{\infty}$ is given by

$$HC^n(A^{\infty}) = \left\{ \begin{array}{cc} \mathbf{C}^3 &: n\,\,\text{even},\ n \geq 2\\ 0 &: n\,\,\text{odd},\ n \geq 1 \end{array} \right.$$

The conjecture asserts that the smooth subalgebra has three-dimensional even cyclic cohomology in degrees at least two and vanishing odd cyclic cohomology in positive degrees. The preceding discussion notes that the known degree-zero cocycles extend from the group ring, but that possible ghost cocycles could make the restriction map non-injective; the conjecture rules out additional cyclic cohomology of this kind.