Quantum symmetries in noncommutative C*-systems

• Autorzy: Marcin Marciniak
• Języki publikacji: EN

Abstrakty

EN

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 𝐴 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 𝐴. It turns out that ω is a KMS state provided that ω is faithful.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1998
• Tom: 43
• Numer: 1
• Strony: 297-307

Daty

wydano

1998

Twórcy

autor

Marcin Marciniak

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

Bibliografia

• [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.