A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres (Q2835239)

The Natsume-Olsen sphere \(C(\mathbb S^{2n-1}_\rho)\) is the universal unital \(C^*\)-algebra generated by normal elements \(z_1,\dots,z_n\) such that \(z_1z_1^*+\ldots+z_nz_n^*=1\) and \(z_kz_j=\rho_{jk}z_jz_k\), where \(\rho_{jk}\in\mathbb C\), \(1\leq j,k\leq n\). Let \(T:C(\mathbb S^{2n-1}_\rho)\to C(\mathbb S^{2n-1}_\rho)\) be the antipodal map given by \(T(z_j)=-z_j\), \(j=1,\dots,n\). The noncommutative Borsuk-Ulam theorem states that an equivariant (for the antipodal maps) unital \(*\)-homomorphism \(\Phi:C(\mathbb S^{2n-1}_\rho)\to C(\mathbb S^{2n-1}_\omega)\) between two Natsume-Olsen spheres induces multiplication by an odd integer on the \(K_1\) groups. This generalizes one of the classical versions of the Borsuk-Ulam theorem: an odd self-map of a sphere is homotopically nontrivial.