Galois groups of quantum group actions and regularity of fixed-point algebras

Yamanouchi, Takehiko

doi:10.1090/S0002-9947-03-03282-3

Galois groups of quantum group actions and regularity of fixed-point algebras
HTML articles powered by AMS MathViewer

by Takehiko Yamanouchi PDF

Trans. Amer. Math. Soc. 355 (2003), 2813-2828 Request permission

Abstract:

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

References

Huzihiro Araki, Daniel Kastler, Masamichi Takesaki, and Rudolf Haag, Extension of KMS states and chemical potential, Comm. Math. Phys. 53 (1977), no. 2, 97–134. MR 443679, DOI 10.1007/BF01609126
Jean De Cannière, On the intrinsic group of a Kac algebra, Proc. London Math. Soc. (3) 40 (1980), no. 1, 1–20. MR 560992, DOI 10.1112/plms/s3-40.1.1
J. Dixmier and O. Maréchal, Vecteurs totalisateurs d’une algèbre de von Neumann, Comm. Math. Phys. 22 (1971), 44–50 (French, with English summary). MR 296708, DOI 10.1007/BF01651583
Michel Enock, Sous-facteurs intermédiaires et groupes quantiques mesurés, J. Operator Theory 42 (1999), no. 2, 305–330 (French, with English summary). MR 1716961
Michel Enock and Jean-Marie Schwartz, Systèmes dynamiques généralisés et correspondances, J. Operator Theory 11 (1984), no. 2, 273–303 (French). MR 749163
Michel Enock and Jean-Marie Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992. With a preface by Alain Connes; With a postface by Adrian Ocneanu. MR 1215933, DOI 10.1007/978-3-662-02813-1
Masaki Izumi, Roberto Longo, and Sorin Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998), no. 1, 25–63. MR 1622812, DOI 10.1006/jfan.1997.3228
Hideki Kosaki, Extension of Jones’ theory on index to arbitrary factors, J. Funct. Anal. 66 (1986), no. 1, 123–140. MR 829381, DOI 10.1016/0022-1236(86)90085-6
Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934 (English, with English and French summaries). MR 1832993, DOI 10.1016/S0012-9593(00)01055-7
J. Kustermans and S. Vaes, Locally compact quantum groups in the von Neumann algebra setting, To appear in Math. Scand..
Mihai Pimsner and Sorin Popa, Entropy and index for subfactors, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 57–106. MR 860811, DOI 10.24033/asens.1504
Jean-Marie Schwartz, Sur la structure des algèbres de Kac. I, J. Functional Analysis 34 (1979), no. 3, 370–406 (French, with English summary). MR 556262, DOI 10.1016/0022-1236(79)90083-1
Şerban Strătilă, Modular theory in operator algebras, Editura Academiei Republicii Socialiste România, Bucharest; Abacus Press, Tunbridge Wells, 1981. Translated from the Romanian by the author. MR 696172
Masamichi Takesaki, The structure of a von Neumann algebra with a homogeneous periodic state, Acta Math. 131 (1973), 79–121. MR 438148, DOI 10.1007/BF02392037
Tamotsu Teruya and Yasuo Watatani, Lattices of intermediate subfactors for type III factors, Arch. Math. (Basel) 68 (1997), no. 6, 454–463. MR 1444657, DOI 10.1007/s000130050078
Stefaan Vaes, The unitary implementation of a locally compact quantum group action, J. Funct. Anal. 180 (2001), no. 2, 426–480. MR 1814995, DOI 10.1006/jfan.2000.3704
Takehiko Yamanouchi, On dominancy of minimal actions of compact Kac algebras and certain automorphisms in $\textrm {Aut}(\scr A/\scr A^\alpha )$, Math. Scand. 84 (1999), no. 2, 297–319. MR 1700534, DOI 10.7146/math.scand.a-13881
Takehiko Yamanouchi, The Connes spectrum for actions of compact Kac algebras and factoriality of their crossed products, Hokkaido Math. J. 28 (1999), no. 2, 409–434. MR 1698434, DOI 10.14492/hokmj/1351001224
Takehiko Yamanouchi, Uniqueness of Haar measures for a quasi Woronowicz algebra, Hokkaido Math. J. 30 (2001), no. 1, 105–112. MR 1815002, DOI 10.14492/hokmj/1350911926
T. Yamanouchi, Description of the automorphism group $\text {Aut}(\mathcal {A}/\mathcal {A}^{\alpha })$ for a minimal action of a compact Kac algebra and its application, J. Funct. Anal., 194 (2002), 1–16.