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

TL;DR

This paper explores the mathematical structure of $oldsymbol{ ext{α-CPD kernels}}$ in $C^*$-algebras, focusing on their Kolmogorov decomposition, Bures distance, and transition probability, advancing the understanding of these kernels in operator algebra theory.

Contribution

It provides a Kolmogorov decomposition for $ ext{α-CPD kernels}$ and characterizes the Bures distance and transition probability for these kernels in the context of $C^*$-algebras.

Findings

Derived the Kolmogorov decomposition for α-CPD kernels

Investigated and characterized the Bures distance between α-CPD kernels

Defined and characterized transition probability for α-CPD kernels

Abstract

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.