Fixed point theorems for completely positive maps in C ⋆ -algebra -valued bipolar b -metric spaces with non-solid cones and applications to quantum mechanics

We develop a fixed point theorem for completely positive maps in C\(^\star\)-algebra-valued bipolar \(b\)-metric spaces over non-solid cones-settings in which the interior of the positivity cone is empty and classical norm-based techniques fail. Our contraction condition replaces the standard scalar inequality \(d(Tx, Ty) \leq k d(x, y)\) with an operator inequality \(\varphi(Tx, Ty) \preceq \Gamma(\varphi(x, y))\), where \(\Gamma\) is a completely positive map satisfying \( < 1\). When \(\Gamma\) is scalar multiplication, our result precisely recovers the Banach contraction principle, showing that it is a special case of our operator-theoretic framework. This approach leverages spectral radius decay and Kraus operator representations to establish existence and uniqueness of fixed points, even in the absence of interior-point structure. Applications include the asymptotic convergence of quantum channels to steady-state density matrices and the stability of operator-based learning models. Our results unify and extend several recent generalizations of fixed point theory, offering a new analytic foundation for contraction behavior in operator-valued and quantum systems.