# Quantum symmetries in noncommutative C*-systems

Banach Center Publications (1998)

- Volume: 43, Issue: 1, page 297-307
- ISSN: 0137-6934

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topMarciniak, Marcin. "Quantum symmetries in noncommutative C*-systems." Banach Center Publications 43.1 (1998): 297-307. <http://eudml.org/doc/208850>.

@article{Marciniak1998,

abstract = {We introduce the notion of a completely quantum C*-system (A,G,α), i.e. a C*-algebra A with an action α of a compact quantum group G. Spectral properties of completely quantum systems are investigated. In particular, it is shown that G-finite elements form the dense *-subalgebra of A. Furthermore, properties of ergodic systems are studied. We prove that there exists a unique α-invariant state ω on A. Its properties are described by a family of modular operators $\{σ_z\}_\{z∈ℂ\}$ acting on . It turns out that ω is a KMS state provided that ω is faithful.},

author = {Marciniak, Marcin},

journal = {Banach Center Publications},

keywords = {spectral properties; completely quantum -system; compact quantum group; ergodic systems; modular operators; KMS state; faithful},

language = {eng},

number = {1},

pages = {297-307},

title = {Quantum symmetries in noncommutative C*-systems},

url = {http://eudml.org/doc/208850},

volume = {43},

year = {1998},

}

TY - JOUR

AU - Marciniak, Marcin

TI - Quantum symmetries in noncommutative C*-systems

JO - Banach Center Publications

PY - 1998

VL - 43

IS - 1

SP - 297

EP - 307

AB - We introduce the notion of a completely quantum C*-system (A,G,α), i.e. a C*-algebra A with an action α of a compact quantum group G. Spectral properties of completely quantum systems are investigated. In particular, it is shown that G-finite elements form the dense *-subalgebra of A. Furthermore, properties of ergodic systems are studied. We prove that there exists a unique α-invariant state ω on A. Its properties are described by a family of modular operators ${σ_z}_{z∈ℂ}$ acting on . It turns out that ω is a KMS state provided that ω is faithful.

LA - eng

KW - spectral properties; completely quantum -system; compact quantum group; ergodic systems; modular operators; KMS state; faithful

UR - http://eudml.org/doc/208850

ER -

## References

top- [1] D. Bernard and G. Felder, Quantum group symmetries in two-dimensional lattice quantum field theory, Nucl. Phys. B 365 (1991), 98-120.
- [2] M. Fannes, B. Nachtergaele and R. F. Werner, Quantum Spin Chains with Quantum Group Symmetry, Commun. Math. Phys. 174 (1996), 477-507. Zbl0845.46043
- [3] K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection Sectors with Braid Group Statistics and Exchange Algebras I, Commun. Math. Phys. 125 (1989), 201-226, II, Rev. Math. Phys. - Special Issue (1992), 113-157. Zbl0682.46051
- [4] R. Haag, Local Quantum Physics, Springer-Verlag, Berlin Heidelberg 1992. Zbl0777.46037
- [5] R. Hοeg-Krohn, M. B. Landstad and E. Stοrmer, Compact ergodic groups of automorphisms, Ann. Math. 114 (1981), 75-86. Zbl0472.46046
- [6] G. Mack and V. Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B 370 (1992), 185-230. Zbl1221.81108
- [7] M. Marciniak, Actions of compact quantum groups on C*-algebras, Proc. Amer. Math. Soc. 126 (1998), 607-616. Zbl0885.22009
- [8] M. Marciniak, Quantum symmetries in noncommutative dynamical systems, Gdańsk University, 1997 (in Polish).
- [9] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London 1979. Zbl0416.46043
- [10] P. Podleś, Symmetries of Quantum Spaces. Subgroups and Quotient Spaces of Quantum SU(2) and SO(3) Groups, Commun. Math. Phys. 170 (1995), 1-20. Zbl0853.46074
- [11] M. E. Sweedler, Hopf algebras, W.A. Benjamin, Inc., New York 1969.
- [12] M. Takesaki, Theory of operator algebras, Springer Verlag, Berlin Heidelberg New York 1979.
- [13] S. L. Woronowicz, Compact Matrix Pseudogroups, Commun. Math. Phys. 111 (1987), 613-665. Zbl0627.58034
- [14] S. L. Woronowicz, Compact quantum groups, preprint 1994.

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