On certain K-groups associated with minimal flows

It is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial K₁-group. We show in this note that if the unique ergodicity is dropped, then such K₁-group can be non-trivial. Therefore, in the general setting of minimal flows, even the K-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.