It is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial $K_1$-group. We show in this note that if the unique ergodicity is dropped, then such $K_1$-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.

MSC Classifications:

46L80, 47B35, 47C15 show english descriptions $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
Operators in $C^*$- or von Neumann algebras 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22]
47B35 - Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]
47C15 - Operators in $C^*$- or von Neumann algebras

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