Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Classification of crossed-product $ C\sp *$-algebras associated with characters on free groups


Author: Hong Sheng Yin
Journal: Trans. Amer. Math. Soc. 320 (1990), 105-143
MSC: Primary 46L55; Secondary 46L80
DOI: https://doi.org/10.1090/S0002-9947-1990-0962286-2
MathSciNet review: 962286
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the classification problem of crossed-product $ {C^ * }$-algebras of the form $ C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf{Z}}$, where $ G$ is a discrete group, $ \chi$ is a one-dimensional character of $ G$, and $ {\alpha_\chi}$ is the unique $ *$-automorphism of $ C_r^ * (G)$ such that if $ U$ is the left regular representation of $ G$, then $ {\alpha_{\chi}(U_{g})=\chi(g)U_{g}}$, $ g \in G$. When $ {G = F_{n}}$, the free group on $ n$ generators, we have a complete classification of these crossed products up to $ *$-isomorphism which generalizes the classification of irrational and rational rotation $ {C^ * }$-algebras. We show that these crossed products are determined by two $ K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $ \operatorname{Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of $ {F_n}$. The classification of crossed products follows from these results.

We also consider the same problem for $ G$ some other groups.


References [Enhancements On Off] (What's this?)

  • [1] C. A. Akemann and P. A. Ostrand, Computing norms in group $ {C^ * }$-algebras, Amer. J. Math. 98 (1976), 1015-1047. MR 0442698 (56:1079)
  • [2] M. F. Atiyah, $ K$-theory, Benjamin, New York, 1967. MR 0224083 (36:7130)
  • [3] M. D. Brabanter, The classification of rational rotation $ {C^ * }$-algebras, Arch. Math. (Basel) 43 (1984), no. 1, 79-83. MR 758343 (86c:46067)
  • [4] M. D. Brabanter and H. Zettl, $ {C^ * }$-algebras associated with rotation groups and characters, Manuscripta Math. 47 (1984), 153-174. MR 744317 (86a:46090)
  • [5] H. Coxeter and W. Moser, Generators and relations for discrete groups, Springer-Verlag, New York, 1972. MR 0349820 (50:2313)
  • [6] J. Cuntz, $ K$-theoretic amenability for discrete groups, J. Reine Angew. Math. 344 (1983), 180-195. MR 716254 (86e:46064)
  • [7] -, The internal structure of simple $ {C^ * }$-algebras, Proc. Sympos. Pure Math., vol. 38, part 1, Amer. Math. Soc., Providence, R.I., 1982. MR 679697 (84h:46072)
  • [8] -, $ K$-theory and $ {C^ * }$-algebras, Lecture Notes in Math., vol. 1046, Springer-Verlag, New York, pp. 55-79. MR 750677 (86d:46071)
  • [9] J. Cuntz, G. Elliott, F. Goodman and P. Jorgensen, On the classification of non-commutative tori, II, C. R. Math. Rep. Acad. Sci Canada 7 (1985), 189-194. MR 789311 (86j:46064b)
  • [10] S. Disney and I. Raeburn, Homogeneous $ {C^ * }$-algebras whose spectra are tori, J. Austral. Math. Soc. (Ser. A) 38 (1985), 9-39. MR 765447 (86i:46057)
  • [11] S. Disney, G. Elliott, A. Kumjian and I. Raeburn, On the classification of non-commutative tori, C. R. Math. Rep. Acad. Sci. Canada 7 (1985), 137-141. MR 781813 (86j:46064a)
  • [12] G. A. Elliott, On the $ K$-theory of the $ {C^ * }$-algebras generated by a projective representation of a torsion-free abelian group, Operator Algebras and Group Representations, vol. I, Pitman, 1984, pp. 157-184. MR 731772 (85m:46067)
  • [13] P. de la Harpe, Reduced $ {C^ * }$-algebras of discrete groups which are simple with unique trace, Lecture Notes in Math, vol. 1132, Springer-Verlag, New York, pp. 230-253.
  • [14] R. Høegh-Krohn and T. Skjelbred, Classification of $ {C^ * }$-algebras admitting ergodic actions of the two-dimensional torus, J. Reine Angew. Math. 328 (1981), 1-8. MR 636190 (83m:46082)
  • [15] S. Itoh, Conditional expectations in $ {C^ * }$-crossed products, Trans. Amer. Math. Soc. 267 (1981), 661-667. MR 626496 (82j:46087)
  • [16] M. Karoubi, $ K$-theory, an introduction, Springer-Verlag, New York, 1976. MR 0488029 (58:7605)
  • [17] E. C. Lance, $ K$-theory for certain group $ {C^ * }$-algebras, Acta Math. 151 (1983), 209-230. MR 723010 (86f:46076)
  • [18] A. W. Mostowski, On automorphisms of relatively free groups, Fund. Math. (1962), 403-411. MR 0137755 (25:1204)
  • [19] D. Olesen, Inner $ *$-automorphisms of simple $ {C^ * }$-algebras, Comm. Math. Phys. 44 (1975), 175-190. MR 0388113 (52:8950)
  • [20] J. A. Packer, $ {C^ * }$-algebras generated by projective representations of the discrete Heisenberg group, J. Operator Theory 18 (1987), 41-66. MR 912812 (89h:46079)
  • [21] W. Paschke, Inner product modules arising from compact automorphism groups of von Neumann algebras, Trans. Amer. Math. Soc. 224 (1976), 87-102. MR 0420294 (54:8308)
  • [22] W. Paschke and N. Salinas, $ {C^ * }$-algebras associated with free products of groups, Pacific J. Math. 82 (1979), 211-221 MR 549845 (82c:22010)
  • [23] G. K. Pedersen, $ {C^ * }$-algebras and their automorphism groups, Academic Press, New York, 1979. MR 548006 (81e:46037)
  • [24] J. Phillips, Automorphisms of full II$ _{1}$ factors, with applications to factors of type III, Duke Math. J. 43 (1976), 375-385. MR 0402518 (53:6337)
  • [25] -, Automorphisms of full II$ _{1}$ factors, Canad. Math. Bull. 21 (1978), 325-328. MR 511580 (80c:46067)
  • [26] M. Pimsner, Ranges of traces on $ {K_0}$ of reduced crossed products by free groups, Lecture Notes in Math., vol. 1132, Springer-Verlag, New York, pp. 374-408. MR 799581 (87j:46131)
  • [27] M. Pimsner and D. Voiculescu, Imbedding the irrational rotation $ {C^ * }$-algebra into an AF-algebra, J. Operator Theory 4 (1980), 201-210. MR 595412 (82d:46086)
  • [28] -, Exact sequences for $ K$-groups and Ext-groups of certain cross-product $ {C^ * }$-algebras, J. Operator Theory 4 (1980), 93-118. MR 587369 (82c:46074)
  • [29] -, $ K$-groups of reduced crossed products by free groups, J. Operator Theory 8 (1982), 131-156. MR 670181 (84d:46092)
  • [30] R. Powers, Simplicity of the $ {C^ * }$-algebra associated with the free group on two generators, Duke Math. J. 42 (1975), 151-156. MR 0374334 (51:10534)
  • [31] N. Riedel, Classification of the $ {C^ * }$-algebras associated with minimal rotations, Pacific J. Math. 101 (1982), 153-161. MR 671848 (84e:46064)
  • [32] M. Rieffel, $ {C^ * }$-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429. MR 623572 (83b:46087)
  • [33] -, The cancellation theorem for projective modules over irrational rotation $ {C^ * }$-algebras, Proc. London Math. Soc. 47 (1983), 285-302. MR 703981 (85g:46085)
  • [34] -, $ K$-theory of crossed products of $ {C^ * }$-algebras by discrete groups, (preprint).
  • [35] J. L. Taylor, Banach algebras and topology, Algebras in Analysis, Academic Press, 1975. MR 0417789 (54:5837)
  • [36] Y. Watatani, Toral automorphisms on irrational rotation algebras, Math. Japon. 26 (1981), 479-484. MR 634924 (82m:46073)
  • [37] H. S. Yin, A simple proof of the classification of rational rotation $ {C^ * }$-algebras, Proc. Amer. Math. Soc. 98 (1986), 469-470. MR 857943 (88a:46062)
  • [38] -, Classification of $ {C^ * }$-crossed products associated with characters on free groups, thesis, Dalhousie University, 1985.
  • [39] G. W. Mackey, Harmonical analysis as the exploitation of symmetry--a historical survey, Bull. Amer. Math. Soc. (N.S.) 3 (1980), 543-698. MR 571370 (81d:01017)
  • [40] P. G. Ghatage and W. J. Phillips, $ {C^ * }$-algebras generated by weighted shifts, II, Indiana Univ. Math. J. 30 (1981), 539-545. MR 620266 (83h:46072)
  • [41] B. Brenken, Representations and automorphisms of the irrational rotation algebra, Pacific J. Math. 111 (1984), 257-282. MR 734854 (86a:46089)
  • [42] H. S. Yin, Classification of $ {C^ * }$-crossed products associated with characters on free groups, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), 1-4. MR 873399 (88b:22005)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L55, 46L80

Retrieve articles in all journals with MSC: 46L55, 46L80


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-0962286-2
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society