Abstract
We characterize the minimal tensor product of two \(C^{\ast}\)-algebras in terms of extension properties of faithful completely positive maps on partial algebras. We discuss connections with the theory of independent quantum subsystems.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
H. Araki, Mathematical Theory of Quantum Fields (Oxford Univ. Press, New York, 2009).
L. J. Bunce and J. Hamhalter, ‘‘\(C^{\ast}\)-independence, product states and commutation,’’ Ann. Henri Poincare 5, 1081–1095 (2004).
R. Conti and J. Hamhalter, ‘‘Independence of group algebras,’’ Math. Nachr. 283, 818–827 (2010).
R. Haag and D. Kastler, ‘‘An algebric approach to quantum field theory,’’ J. Math. Phys. 5, 848–861 (1964).
J. Hamhalter and S. **, ‘‘Operational independence and tensor products of \(C^{\ast}\)-algebra,’’ J. Math. Phys. 58 (2017).
M. Florig and S. J. Summers, ‘‘On the statistical independence of algebras of observables,’’ J. Math. Phys. 38, 1318–1328 (1997).
J. Hamhalter, ‘‘Statistical independence of operator algebras,’’ Ann. Inst. Henri Poincaré 67, 447–462 (1997).
J. Hamhalter, ‘‘\(C^{\ast}\)-independence and \(W^{\ast}\)-independence of von Neumann algebras,’’ Math. Nachr. 239–240, 146–156 (2002).
J. Hamhalter, Quantum Measure Theory (Kluwer Academic, Dordrecht, 2003).
K. Kraus, States, Effects, and Operations (Springer, Berlin, 1983).
G. Pisier, Tensor Products of \(C^{\ast}\) -Algebras and Operator Spaces: The Conners–Kirchberg Problem, Vol. 96 of London Mathematical Society Student Texts (London Math. Soc., London, 2020).
M. Redei, Quantum Logic in Algebraic Approach, Vol. 91 of Fundamental Theories of Physics (Kluwer Academic, Dordrecht, 1998).
M. Redei and S. J. Summers, ‘‘When are quantum systems operationally independent?,’’ Int. J. Theor. Phys. 49, 3250–3261 (2010).
M. Redei, ‘‘Operational separability and operational independence in algebraic quantum mechanics,’’ Found. Phys. 40, 1439–1449 (2010).
H. Roos, ‘‘Independence of local algebras in quantum field theory,’’ Commun. Math. Phys. 37, 273–286 (1974).
S. J. Summers, ‘‘On the independence of local algebras in quantum field theory,’’ Rev. Math. Phys. 2, 201–247 (1990).
M. Takesaki, Theory of Operator Algebras, I (Springer, New York, 2002).
Funding
The work of Jan Hamhalter was supported by the grants: GA23-04776S ‘‘Interplay of algebraic, metric, geometric and topological structures on Banach spaces’’ and SGS22/053/OHK3/1T/13 ‘‘Methods of modern mathematics and its applications.’’
Author information
Authors and Affiliations
Corresponding authors
Additional information
(Submitted by A. M. Elizarov)
Rights and permissions
About this article
Cite this article
Hamhalter, J., Turilova, E. Operational Independence, Faithful Maps and Minimal Tensor Products. Lobachevskii J Math 44, 2027–2032 (2023). https://doi.org/10.1134/S1995080223060203
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1995080223060203