Fixed point theorems for completely positive maps in \(C^\star\text{-algebra}\)-valued bipolar \(b\)-metric spaces with non-solid cones and applications to quantum mechanics
Authors
A. T. Bokodisa
- Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria, South Africa.
A. Aphane
- Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria, South Africa.
Abstract
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.
Share and Cite
ISRP Style
A. T. Bokodisa, A. Aphane, Fixed point theorems for completely positive maps in \(C^\star\text{-algebra}\)-valued bipolar \(b\)-metric spaces with non-solid cones and applications to quantum mechanics, Journal of Nonlinear Sciences and Applications, 18 (2025), no. 4, 259--271
AMA Style
Bokodisa A. T., Aphane A., Fixed point theorems for completely positive maps in \(C^\star\text{-algebra}\)-valued bipolar \(b\)-metric spaces with non-solid cones and applications to quantum mechanics. J. Nonlinear Sci. Appl. (2025); 18(4):259--271
Chicago/Turabian Style
Bokodisa, A. T., Aphane, A.. "Fixed point theorems for completely positive maps in \(C^\star\text{-algebra}\)-valued bipolar \(b\)-metric spaces with non-solid cones and applications to quantum mechanics." Journal of Nonlinear Sciences and Applications, 18, no. 4 (2025): 259--271
Keywords
- \(C^\star\text{-algebra}\)
- quantum mechanics
- operator-based models
- completely positive maps
- non-solid cones
- fixed point theory
- bipolar b-metric spaces
MSC
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