Download PDF - Quantum symmetries in noncommutative C*-systemsAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Quantum symmetries in noncommutative C*-systems

Authors 1

Affiliations

  1. Institute of Mathematics, Gdańsk University, Wita Stwosza 57, 80-952 Gdańsk, Poland

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 {σz}z∈ℂ acting on . It turns out that ω is a KMS state provided that ω is faithful.

Bibliography

  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.
  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.
  4. R. Haag, Local Quantum Physics, Springer-Verlag, Berlin Heidelberg 1992.
  5. R. Hοeg-Krohn, M. B. Landstad and E. Stοrmer, Compact ergodic groups of automorphisms, Ann. Math. 114 (1981), 75-86.
  6. G. Mack and V. Schomerus, Quasi Hopf quantum symmetry in quantum theory, Nucl. Phys. B 370 (1992), 185-230.
  7. M. Marciniak, Actions of compact quantum groups on C*-algebras, Proc. Amer. Math. Soc. 126 (1998), 607-616.
  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.
  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.
  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.
  14. S. L. Woronowicz, Compact quantum groups, preprint 1994.
Banach Center Publications
1998/43/1
Export to PBN, Opens in new tab
Pages:
297-307
Main language of publication
English
Published
1998
Exact and natural sciences