Smooth crossed products induced by minimal unique ergodic diffeomorphisms on odd spheres

TL;DR

The paper demonstrates that while certain $C^*$-crossed product algebras are isomorphic for different spheres, their smooth counterparts are distinguished by cyclic cohomology, revealing finer invariants.

Contribution

It shows that smooth crossed product algebras induced by minimal unique ergodic diffeomorphisms on odd spheres are not isomorphic when the sphere dimensions differ, despite $C^*$-algebra isomorphisms.

Findings

$C^*$-crossed products are isomorphic for different sphere dimensions.

Smooth crossed products are not isomorphic when sphere dimensions differ.

Cyclic cohomology distinguishes smooth crossed products by sphere dimension.

Abstract

For minimal unique ergodic diffeomorphisms $\alpha_n$ of $S^{2n+1} (n>0)$ and $\alpha_m$ of $S^{2m+1}(m>0)$, the $C^*$-crossed product algebra $C(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z}$ is isomorphic to $C(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z}$ even though $n\neq m$ . However, by cyclic cohomology, we show that smooth crossed product algebra $C^\infty(S^{2n+1})\rtimes_{\alpha_n} \mathbb{Z}$ is not isomorphic to $C^\infty(S^{2m+1})\rtimes_{\alpha_m} \mathbb{Z}$ if $n\neq m$.