  If the symmetry (fixed invertible self adjoint map) of Krein spaces is
replaced by a fixed unitary, then we obtain the notion of S-spaces which was
introduced by Szafraniec. Assume $\alpha$ to be an automorphism on a
$C^*$-algebra. In this article, we obtain the Kolmogorov decomposition of
$\alpha$-completely positive definite (or $\alpha$-CPD-kernels for short) and
investigate the Bures distance between $\alpha$-CPD-kernels. We also define
transition probability for these kernels and find a characterization of the
transition probability.
