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Acta Applicandae Mathematicae

, Volume 108, Issue 3, pp 585–602 | Cite as

Commutativity and Ideals in Strongly Graded Rings

  • Johan Öinert
  • Sergei Silvestrov
  • Theodora Theohari-Apostolidi
  • Harilaos Vavatsoulas
Article

Abstract

In some recent papers by the first two authors it was shown that for any algebraic crossed product \(\mathcal {A}\) , where \(\mathcal {A}_{0}\) , the subring in the degree zero component of the grading, is a commutative ring, each non-zero two-sided ideal in \(\mathcal {A}\) has a non-zero intersection with the commutant \(C_{\mathcal {A}}(\mathcal {A}_{0})\) of \(\mathcal {A}_{0}\) in \(\mathcal {A}\) . This result has also been generalized to crystalline graded rings; a more general class of graded rings to which algebraic crossed products belong. In this paper we generalize this result in another direction, namely to strongly graded rings (in some literature referred to as generalized crossed products) where the subring \(\mathcal {A}_{0}\) , the degree zero component of the grading, is a commutative ring. We also give a description of the intersection between arbitrary ideals and commutants to bigger subrings than \(\mathcal {A}_{0}\) , and this is done both for strongly graded rings and crystalline graded rings.

Keywords

Strongly graded rings Commutativity Ideals 

Mathematics Subject Classification (2000)

16W50 13A02 16D25 46H10 

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References

  1. 1.
    an Huef, A., Raeburn, I.: The ideal structure of Cuntz-Krieger algebras. Ergod. Theory Dyn. Syst. 17(3), 611–624 (1997) MATHCrossRefGoogle Scholar
  2. 2.
    Archbold, R.J., Spielberg, J.S.: Topologically free actions and ideals in discrete C *-dynamical systems. Proc. Edinb. Math. Soc. (2) 37(1), 119–124 (1994) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bratteli, O., Jorgensen, P.E.T.: Iterated function systems and permutation representations of the Cuntz algebra. Mem. Am. Math. Soc. 139(663), x+89p. (1999) MathSciNetGoogle Scholar
  4. 4.
    Bratteli, O., Jorgensen, P.: Wavelets through a looking glass. In: The World of the Spectrum. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2002). xxii+389 p. Google Scholar
  5. 5.
    Bratteli, O., Evans, D.E., Jorgensen, P.E.T.: Compactly supported wavelets and representations of the Cuntz relations. Appl. Comput. Harmon. Anal. 8(2), 166–196 (2000) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Caenepeel, S., Van Oystaeyen, F.: Brauer Groups and the Cohomology of Graded Rings. Monographs and Textbooks in Pure and Applied Mathematics, vol. 121. Marcel Dekker, New York (1988). xii+261 p. MATHGoogle Scholar
  7. 7.
    Carlsen, T.M., Silvestrov, S.: C *-crossed products and shift spaces. Expo. Math. 25(4), 275–307 (2007) MATHMathSciNetGoogle Scholar
  8. 8.
    Cohen, M., Montgomery, S.: Group-graded rings, smash products and group actions. Trans. Am. Math. Soc. 282(1), 237–258 (1984) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dai, X., Larson, D.R.: Wandering vectors for unitary systems and orthogonal wavelets. Mem. Am. Math. Soc. 134(640), vii+68 p. (1998) MathSciNetGoogle Scholar
  10. 10.
    Davidson, K.R.: C *-Algebras by Example. Fields Institute Monographs, vol. 6. American Mathematical Society, Providence (1996) Google Scholar
  11. 11.
    Dutkay, D.E., Jorgensen, P.E.T.: Martingales, endomorphisms, and covariant systems of operators in Hilbert space. J. Oper. Theory 58(2), 269–310 (2007) MATHMathSciNetGoogle Scholar
  12. 12.
    Exel, R.: Crossed-products by finite index endomorphisms and KMS states. J. Funct. Anal. 199(1), 153–188 (2003) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Exel, R.: A new look at the crossed-product of a C *-algebra by an endomorphism. Ergod. Theory Dyn. Syst. 23(6), 1733–1750 (2003) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Exel, R., Vershik, A.: C *-algebras of irreversible dynamical systems. Can. J. Math. 58(1), 39–63 (2006) MATHMathSciNetGoogle Scholar
  15. 15.
    Fisher, J.W., Montgomery, S.: Semiprime skew group rings. J. Algebra 52(1), 241–247 (1978) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Formanek, E., Lichtman, A.I.: Ideals in group rings of free products. Israel J. Math. 31(1), 101–104 (1978) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Irving, R.S.: Prime ideals of Ore extensions over commutative rings. J. Algebra 56(2), 315–342 (1979) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Irving, R.S.: Prime ideals of Ore extensions over commutative rings II. J. Algebra 58(2), 399–423 (1979) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Jørgensen, P.E.T.: Operators and Representation Theory, Canonical Models for Algebras of Operators Arising in Quantum Mechanics. North-Holland Mathematics Studies, vol. 147. North-Holland, Amsterdam (1988). viii+337 p. Google Scholar
  20. 20.
    Jorgensen, P.E.T.: Analysis and Probability: Wavelets, Signals, Fractals. Graduate Texts in Mathematics, vol. 234. Springer, New York (2006). xlviii+276 p. MATHGoogle Scholar
  21. 21.
    Kajiwara, T., Watatani, Y.: C *-algebras associated with complex dynamical systems. Indiana Univ. Math. J. 54(3), 755–778 (2005) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Karpilovsky, G.: The Algebraic Structure of Crossed Products. North-Holland Mathematics Studies, vol. 142. North-Holland, Amsterdam (1987). x+348 p. MATHCrossRefGoogle Scholar
  23. 23.
    Launois, S., Lenagan, T.H., Rigal, L.: Quantum unique factorisation domains. J. Lond. Math. Soc. (2) 74(2), 321–340 (2006) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Leroy, A., Matczuk, J.: Primitivity of skew polynomial and skew Laurent polynomial rings. Commun. Algebra 24(7), 2271–2284 (1996) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Lorenz, M., Passman, D.S.: Centers and prime ideals in group algebras of polycyclic-by-finite groups. J. Algebra 57(2), 355–386 (1979) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Lorenz, M., Passman, D.S.: Prime ideals in crossed products of finite groups. Israel J. Math. 33(2), 89–132 (1979) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lorenz, M., Passman, D.S.: Addendum—prime ideals in crossed products of finite groups. Israel J. Math. 35(4), 311–322 (1980) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Mackey, G.W.: Induced Representations of Groups and Quantum Mechanics. Benjamin, New York/Amsterdam (1968). viii+167 p. MATHGoogle Scholar
  29. 29.
    Mackey, G.W.: The Theory of Unitary Group Representations. Chicago Lectures in Mathematics. University of Chicago Press, Chicago/London (1976). x+372 p. MATHGoogle Scholar
  30. 30.
    Mackey, G.W.: Unitary Group Representations in Physics, Probability, and Number Theory, 2nd edn. Advanced Book Classics. Addison-Wesley Advanced Book Program, Redwood City (1989). xxviii+402 p. MATHGoogle Scholar
  31. 31.
    Marubayashi, H., Nauwelaerts, E., Van Oystaeyen, F.: Graded rings over arithmetical orders. Commun. Algebra 12(5–6), 745–775 (1984) MATHCrossRefGoogle Scholar
  32. 32.
    Montgomery, S., Passman, D.S.: Crossed products over prime rings. Israel J. Math. 31(3–4), 224–256 (1978) MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Nǎstǎsescu, C., Van Oystaeyen, F.: Methods of Graded Rings. Lecture Notes in Mathematics, vol. 1836. Springer, Berlin (2004). xiv+304 p. Google Scholar
  34. 34.
    Nauwelaerts, E., Van Oystaeyen, F.: Introducing crystalline graded algebras. Algebr. Represent. Theory 11(2), 133–148 (2008) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Neijens, T., Van Oystaeyen, F., Yu, W.W.: Centers of certain crystalline graded rings. Preprint in preparation (2007) Google Scholar
  36. 36.
    Öinert, J., Silvestrov, S.D.: Commutativity and ideals in algebraic crossed products. J. Gen. Lie. T. Appl. 2(4), 287–302 (2008) MATHCrossRefGoogle Scholar
  37. 37.
    Öinert, J., Silvestrov, S.D.: On a correspondence between ideals and commutativity in algebraic crossed products. J. Gen. Lie. T. Appl. 2(3), 216–220 (2008) MATHGoogle Scholar
  38. 38.
    Öinert, J., Silvestrov, S.D.: Crossed product-like and pre-crystalline graded rings. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds.) Generalized Lie Theory in Mathematics, Physics and Beyond, Chap. 24, pp. 281–296. Springer, Berlin/Heidelberg (2008) Google Scholar
  39. 39.
    Ostrovskyĭ, V., Samoĭlenko, Y.: Introduction to the Theory of Representations of Finitely Presented *-Algebras. I. Representations by Bounded Operators. Reviews in Mathematics and Mathematical Physics, vol. 11, p. 1. Harwood Academic, Amsterdam (1999). iv+261 p. Google Scholar
  40. 40.
    Passman, D.S.: Infinite Crossed Products. Pure and Applied Mathematics, vol. 135. Academic Press, Boston (1989) MATHGoogle Scholar
  41. 41.
    Passman, D.S.: The Algebraic Structure of Group Rings. Pure and Applied Mathematics. Wiley-Interscience, New York (1977). xiv+720 p. MATHGoogle Scholar
  42. 42.
    Pedersen, G.K.: C *-Algebras and Their Automorphism Groups. London Mathematical Society Monographs, vol. 14. Academic Press, London/New York (1979) MATHGoogle Scholar
  43. 43.
    Rowen, L.: Some results on the center of a ring with polynomial identity. Bull. Am. Math. Soc. 79, 219–223 (1973) MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Silvestrov, S.D., Tomiyama, J.: Topological dynamical systems of type I. Expo. Math. 20(2), 117–142 (2002) MATHMathSciNetGoogle Scholar
  45. 45.
    Svensson, C., Silvestrov, S., de Jeu, M.: Dynamical systems and commutants in crossed products. Int. J. Math. 18(4), 455–471 (2007) MATHCrossRefGoogle Scholar
  46. 46.
    Svensson, C., Silvestrov, S., de Jeu, M.: Connections between dynamical systems and crossed products of Banach algebras by ℤ. In: Janas, J., Kurasov, P., Laptev, A., Naboko, S., Stolz, G. (eds.) Methods of Spectral Analysis in Mathematical Physics. Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2006. Lund, Sweden, Operator Theory: Advances and Applications, vol. 186, pp. 391–401. Birkhäuser, Basel (2009) Google Scholar
  47. 47.
    Svensson, C., Silvestrov, S., de Jeu, M.: Dynamical systems associated to crossed products, Preprints in Mathematical Sciences 2007:22, LUFTMA-5088-2007. Leiden Mathematical Institute report 2007-30. arxiv:0707.1881. To appear in Acta Appl. Math.
  48. 48.
    Svensson, C., Tomiyama, J.: On the commutant of C(X) in C *-crossed products by ℤ and their representations. arXiv:0807.2940
  49. 49.
    Theohari-Apostolidi, T., Vavatsoulas, H.: On strongly group graded algebras and orders. New techniques in Hopf algebras and graded ring theory, pp. 179–186. K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels (2007) Google Scholar
  50. 50.
    Theohari-Apostolidi, T., Vavatsoulas, H.: Induced modules of strongly group-graded algebras. Colloq. Math. 108(1), 93–104 (2007) MATHCrossRefMathSciNetGoogle Scholar
  51. 51.
    Theohari-Apostolidi, T., Vavatsoulas, H.: On strongly graded Gorestein orders. Algebra Discrete Math. 2005(2), 80–89 (2005) MATHMathSciNetGoogle Scholar
  52. 52.
    Theohari-Apostolidi, T., Vavatsoulas, H.: On the separability of the restriction functor. Algebra Discrete Math. 2003(3), 95–101 (2003) MATHMathSciNetGoogle Scholar
  53. 53.
    Theohari-Apostolidi, T., Vavatsoulas, H.: Induction and Auslander-Reiten sequences over crossed products. In: Formal Power Series and Algebraic Combinatorics, pp. 765–774. Springer, Berlin (2000) Google Scholar
  54. 54.
    Theohari-Apostolidi, T., Vavatsoulas, H.: On induced modules over strongly group-graded algebras. Beiträge Algebra Geom. 40(2), 373–383 (1999) MATHMathSciNetGoogle Scholar
  55. 55.
    Tomiyama, J.: Invitation to C *-Algebras and Topological Dynamics. World Scientific Advanced Series in Dynamical Systems, vol. 3. World Scientific, Singapore (1987). x+167 p. Google Scholar
  56. 56.
    Tomiyama, J.: The Interplay Between Topological Dynamics and Theory of C *-Algebras. Lecture Notes Series, vol. 2. Seoul National University Research Institute of Mathematics Global Anal Research Center, Seoul (1992). vi+69 p. MATHGoogle Scholar
  57. 57.
    Tomiyama, J.: The interplay between topological dynamics and theory of C *-algebras II (after the Seoul lecture note 1992). Sūrikaisekikenkyūsho Kōkyūroku 1151, 1–71 (2000) MATHGoogle Scholar
  58. 58.
    Williams, D.P.: Crossed Products of C *-Algebras. Mathematical Surveys and Monographs, vol. 134. American Mathematical Society, Providence (2007). xvi+528 p. MATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Johan Öinert
    • 1
  • Sergei Silvestrov
    • 1
  • Theodora Theohari-Apostolidi
    • 2
  • Harilaos Vavatsoulas
    • 2
  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden
  2. 2.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece

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