Bures distance and transition probability for $\alpha$-CPD-kernels

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.