Stability of Rotation Relations of Three Unitaries with the Flip Action in C*-Algebras

Abstract

The authors show that if Θ = (θ _jk) is a 3 × 3 totally irrational real skew-symmetric matrix, where θ _jk ∈ [0, 1) for j, k = 1, 2, 3, then for any ε > 0, there exists δ > 0 satisfying the following: For any unital C*-algebra A with the cancellation property, strict comparison and nonempty tracial state space, any four unitaries u ₁, u ₂, u ₃,w ∈ A such that (1) $||{u_k}{u_j} - {{\rm{e}}^{2\pi {\rm{i}}{\theta _{jk}}}}{u_j}{u_k}||\, < \delta $ wu _j w ⁻¹ = u _j ⁻¹, w ² = 1 _A for j, k = 1, 2, 3; (2) τ(aw) = 0 and $\tau ({({u_k}{u_j}u_k^ * u_j^ * )^n}) = {{\rm{e}}^{^{2\pi {\rm{i}}n{\theta _{jk}}}}}$ for all n ∈ ℕ, all a ∈ C*(u ₁, u ₂, u ₃), j, k = 1, 2, 3 and all tracial states τ on A, where C*(u ₁, u ₂, u ₃) is the C*-subalgebra generated by u ₁, u ₂ and u ₃, there exists a 4-tuple of unitaries ${\tilde u_1},{\tilde u_2},{\tilde u_3},\tilde w$ in A such that ${\tilde u_k}{\tilde u_j} = {{\rm{e}}^{2\pi {\rm{i}}{\theta _{jk}}}}{\tilde u_j}{\tilde u_k},\,\,\,\,\,\,\,\tilde w{\tilde u_j}{\tilde w^{ - 1}} = \tilde u_j^{ - 1},\,\,\,\,\,{\tilde w^2} = {1_A}\,\,\,\,$ and $\Vert{u_j} - {\tilde u_j}\Vert < \varepsilon ,\,\,\,\,\Vert w - \tilde w\Vert < \varepsilon $ for j, k = 1, 2, 3. The above conclusion is also called that the rotation relations of three unitaries with the flip action is stable under the above conditions.

Keywords

C*-Algebras / Stability / Rotation relation / Unitary / Flip action

References

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