Quantum symmetries in noncommutative C*-systems

Marcin Marciniak

Banach Center Publications (1998)

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

Abstract

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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.

How to cite

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Marciniak, 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

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  8. [8] M. Marciniak, Quantum symmetries in noncommutative dynamical systems, Gdańsk University, 1997 (in Polish). 
  9. [9] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London 1979. Zbl0416.46043
  10. [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. [11] M. E. Sweedler, Hopf algebras, W.A. Benjamin, Inc., New York 1969. 
  12. [12] M. Takesaki, Theory of operator algebras, Springer Verlag, Berlin Heidelberg New York 1979. 
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