Equality of commutator and semicommutator ideals in Toeplitz algebras of minimal flows

Determine whether, for every minimal ergodic flow α on a separable compact Hausdorff space X, the commutator ideal of the Toeplitz algebra T(X, α) equals the semi-commutator ideal SC(X, α); equivalently, prove equality holds for all minimal ergodic flows or construct a minimal ergodic flow (X, α) for which SC(X, α) properly contains the commutator ideal.

Background

The paper studies Toeplitz operators associated with a minimal ergodic flow α on a compact space X and the resulting Toeplitz C*-algebra T(X, α). The semicommutator ideal SC(X, α) is generated by elements of the form T_φ T_ψ − T_{φψ}. The authors note that the commutator ideal is contained in SC(X, α).

A result from Curto, Muhly, and Xia [5, Lemma 24.1] shows that if the action is strictly ergodic, then the commutator and semicommutator ideals coincide. The author remarks that it might be true more generally for minimal ergodic flows, but no example is known where they differ, and the issue is not pursued further in the paper.