  We give a new proof to show that the complex interpolation for families of
Banach spaces is not stable under rearrangement of the given family on the
boundary, although, by a result due to Coifman, Cwikel, Rochberg, Sagher and
Weiss, it is stable when the latter family takes only 2 values. The
non-stability for families taking 3 values was first obtained by Cwikel and
Janson. Our method links this problem to the theory of matrix-valued Toeplitz
operator and we are able to characterize all the transformations on $\T$ that
are invariant for complex interpolation at 0, they are precisely the
origin-preserving inner functions.
