Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 060, 18 pages      arXiv:2111.13076      https://doi.org/10.3842/SIGMA.2022.060

The Gauge Group and Perturbation Semigroup of an Operator System

Rui Dong
Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

Received December 01, 2021, in final form July 28, 2022; Published online August 09, 2022

Abstract
The perturbation semigroup was first defined in the case of $*$-algebras by Chamseddine, Connes and van Suijlekom. In this paper, we take $\mathcal{E}$ as a concrete operator system with unit. We first give a definition of gauge group $\mathcal{G}(\mathcal{E})$ of $\mathcal{E}$, after that we give the definition of perturbation semigroup of $\mathcal{E}$, and the closed perturbation semigroup of $\mathcal{E}$ with respect to the Haagerup tensor norm. We also show that there is a continuous semigroup homomorphism from the closed perturbation semigroup to the collection of unital completely bounded Hermitian maps over $\mathcal{E}$. Finally we compute the gauge group and perturbation semigroup of the Toeplitz system as an example.

Key words: operator algebras; operator systems; functional analysis; noncommutative geometry.

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References

1. Arveson W.B., Subalgebras of $C^{\ast} $-algebras, Acta Math. 123 (1969), 141-224.
2. Blecher D.P., Le Merdy C., Operator algebras and their modules - an operator space approach, London Mathematical Society Monographs. New Series, Vol. 30, Oxford University Press, Oxford, 2004.
3. Chamseddine A.H., Connes A., van Suijlekom W.D., Inner fluctuations in noncommutative geometry without the first order condition, J. Geom. Phys. 73 (2013), 222-234, arXiv:1304.7583.
4. Connes A., van Suijlekom W.D., Spectral truncations in noncommutative geometry and operator systems, Comm. Math. Phys. 383 (2021), 2021-2067, arXiv:2004.14115.
5. Effros E.G., Kishimoto A., Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36 (1987), 257-276.
6. Effros E.G., Ruan Z.-J., Operator spaces, London Mathematical Society Monographs. New Series, Vol. 23, The Clarendon Press, Oxford University Press, New York, 2000.
7. Farenick D., The operator system of Toeplitz matrices, Trans. Amer. Math. Soc. Ser. B 8 (2021), 999-1023, arXiv:2103.16546.
8. Hesp L., The perturbation semigroup of $C^*$-algebras, Master's Thesis, Radboud University Nijmegen, 2016.
9. Kraus K., General state changes in quantum theory, Ann. Physics 64 (1971), 311-335.
10. Neumann N., van Suijlekom W.D., Perturbation semigroup of matrix algebras, J. Noncommut. Geom. 10 (2016), 245-264, arXiv:1410.5961.
11. Paulsen V., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, Vol. 78, Cambridge University Press, Cambridge, 2002.
12. Pisier G., Introduction to operator space theory, London Mathematical Society Lecture Note Series, Vol. 294, Cambridge University Press, Cambridge, 2003.
13. Poluikis J.A., Hill R.D., Completely positive and Hermitian-preserving linear transformations, Linear Algebra Appl. 35 (1981), 1-10.
14. ter Horst S., van der Merwe A., Hill representations for $*$-linear matrix maps, Indag. Math. (N.S.) 33 (2022), 334-356, arXiv:2103.14500.
15. van Suijlekom W.D., Noncommutative geometry and particle physics, Mathematical Physics Studies, Springer, Dordrecht, 2015.