K-theory rank conjecture for noncommutative torus orbifold crossed products

About 10 years old · traced to

Let Aθalg\mathcal A_\theta^{alg} be the algebraic noncommutative torus and let Γ\Gamma be one of the cyclic groups Z3\mathbb Z_3, Z4\mathbb Z_4, or Z6\mathbb Z_6. The group K0(Aθalg⋊Γ)K_0(\mathcal A_\theta^{alg} \rtimes \Gamma) denotes the zeroth K-theory group of the crossed product. K-theory rank conjecture.

K0(Aθalg⋊Γ)≅{Z7for Γ=Z3,Z8for Γ=Z4,Z9for Γ=Z6.K_0(\mathcal A_\theta^{alg} \rtimes \Gamma) \cong \begin{cases} \mathbb Z^7 & \text{for } \Gamma=\mathbb Z_3, \\ \mathbb Z^8 & \text{for } \Gamma=\mathbb Z_4, \\ \mathbb Z^9 & \text{for } \Gamma=\mathbb Z_6. \end{cases}

This conjecture predicts the ranks of the zeroth K-theory groups for the three cyclic orbifold actions considered in the paper. The source gives no resolution status.

References

Primary source

Safdar Quddus, “Cyclic Cohomology and Chern Connes pairing of some crossed product algebras”, arXiv:1605.04551 (2017).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.