Fundamental Group of Simple C*-algebras with Unique Trace III

Nawata, Norio

Canadian journal of mathematics, Tome 64 (2012) no. 3, pp. 573-587

Résumé

We introduce the fundamental group $\mathcal{F}\left( A \right)$ of a simple $\sigma $ -unital ${{C}^{*}}$ –algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple ${{C}^{*}}$ -algebras with Unique Trace I and II by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless ${{C}^{*}}$ -algebras. We show that there exist separable stably projectionless ${{C}^{*}}$ -algebras such that their fundamental groups are equal to $\mathbb{R}_{+}^{\times }$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Détail

DOI : 10.4153/CJM-2011-052-0

Mots-clés : 46L05, 46L08, 46L35, fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension function

Bibliographie

[1] [1] Blackadar, B., Operator algebras: Theory of C*-algebras and von Neumann algebras. Encyclopaedia of Mathematical Sciences, 122, Operator Algebras and Non-commutative Geometry, III, Springer-Verlag, Berlin, 2006. Google Scholar

[2] [2] Blackadar, B. and Handelman, D., Dimension functions and traces on C*-algebras. J. Funct. Anal. 45(1982), no. 3, 297–340. Google Scholar | DOI

[3] [3] Blackadar, B. and Kumjian, A., Skew products of relations and the structure of simple C*-algebras. Math. Z. 189(1985), no. 1, 55–63. Google Scholar | DOI

[4] [4] Brown, L. G., Stable isomorphism of hereditary subalgebras of C*-algebras. Pacific J. Math. 71(1977), no. 2, 335–348. Google Scholar

[5] [5] Brown, L. G., Green, P., and Rieffel, M. A., Stable isomorphism and strong Morita equivalence of C*-algebras. Pacific J. Math. 71(1977), no. 2, 349–363. Google Scholar

[6] [6] Combes, F. and Zettl, H., Order structures, traces and weights on Morita equivalent C*-algebras. Math. Ann. 265(1983), no. 1, 67–81. Google Scholar | DOI

[7] [7] Connes, A., A factor of type II1 with countable fundamental group. J. Operator Theory 4(1980), no. 1, 151–153. Google Scholar

[8] [8] Cuntz, J., Dimension functions on simple C*-algebras. Math. Ann. 233(1978), no. 2, 145–153. Google Scholar | DOI

[9] [9] Cuntz, J. and Pedersen, G. K., Equivalence and traces on C*-algebras. J. Funct. Anal. 33(1979), no. 2, 135–164. Google Scholar | DOI

[10] [10] Elliott, G. A., An invariant for simple C*-algebras. Canadian Mathematical Society. 1945–1995, Vol. 3, Canadian Math. Soc., Ottawa, ON, 1996, pp. 61–90. Google Scholar

[11] [11] Elliott, G. A. and Villadsen, J., Perforated ordered K0-groups. Canad. J. Math. 52(2000), no. 6, 1164–1191. Google Scholar | DOI

[12] [12] Frank, M. and Larson, D., A module frame concept for Hilbert C*-modules. In: The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), Contemp. Math., 247, American Mathematical Society, Providence, RI, 1999, pp. 207–233. Google Scholar

[13] [13] Frank, M. and Larson, D., Frames in Hilbert C*-modules and C*-algebras. J. Operator Theory 48(2002), no. 2, 273–314. Google Scholar

[14] [14] Izumi, M., Kajiwara, T., and Watatani, Y., KMS states and branched points. Ergodic Theory Dynam. Systems 27(2007), no. 6, 1887–1918. Google Scholar

[15] [15] Jacelon, B., A simple self-absorbing, stably projectionless C*-algebra. arxiv:1006.5397v1 Google Scholar

[16] [16] Kajiwara, T., Pinzari, C., and Watatani, Y., Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules. J. Funct. Anal. 159(1998), no. 2, 295–322. Google Scholar | DOI

[17] [17] Kajiwara, T., Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory. J. Funct. Anal. 215(2004), no. 1, 1–49. Google Scholar | DOI

[18] [18] Kajiwara, T. and Watatani, Y., Jones index theory by Hilbert C*-bimodules and K-theory. Trans. Amer. Math. Soc. 352(2000), no. 8, 3429–3472. Google Scholar | DOI

[19] [19] Kasparov, G. G., Hilbert C*-modules: theorems of Stinespring and Voiculescu. J. Operator Theory 4(1980), no. 1, 133–150. Google Scholar

[20] [20] Kishimoto, A. and Kumjian, A., Simple stably projectionless C*-algebras arising as crossed products. Canad. J. Math. 48(1996), no. 5, 980–996. Google Scholar | DOI

[21] [21] Kishimoto, A. and Kumjian, A., Crossed products of Cuntz algebras by quasi-free automorphisms. In: Operator algebras and their applications (Waterloo, ON, 1994/1995), Fields Inst. Commun., 13, American Mathematical Society, Providence, RI, 1997, pp. 173–192. Google Scholar

[22] [22] Kodaka, K., Full projections, equivalence bimodules and automorphisms of stable algebras of unital C*-algebras. Operator Theory, J., 37(1997), no. 2, 357–369. Google Scholar

[23] [23] Kodaka, K., Picard groups of irrational rotation C*-algebras. J. London Math. Soc. (2) 56(1997), no. 1, 179–188. Google Scholar | DOI

[24] [24] Kodaka, K., Projections inducing automorphisms of stable UHF-algebras. Glasg. Math. J. 41(1999), no. 3, 345–354. Google Scholar | DOI

[25] [25] Laca, M. and Neshveyev, S., KMS states of quasi-free dynamics on Pimsner algebras. J. Funct. Anal. 211(2004), no. 2, 457–482. Google Scholar | DOI

[26] [26] Lance, E. C., Hilbert C*-modules. A toolkit for operator algebraists. London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995. Google Scholar

[27] [27] Manuilov, V. M. and Troitsky, E. V., Hilbert C*-modules. Translations of Mathematical Monographs, 226, American Mathematical Society, Providence, RI, 2005. Google Scholar

[28] [28] Murray, F. J. and von Neumann, J., On rings of operators. IV. Ann. of Math. 44(1943), 716–808. Google Scholar | DOI

[29] [29] Nawata, N. and Watatani, Y., Fundamental group of simple C*-algebras with unique trace. Adv. Math. 225(2010), no. 1, 307–318. Google Scholar | DOI

[30] [30] Nawata, N. and Watatani, Y., Fundamental group of simple C*-algebras with unique trace II. J. Funct. Anal. 260(2011), no. 2, 428–435. Google Scholar | DOI

[31] [31] Pedersen, G. K., Measure theory for C-algebras. III. Math. Scand. 25(1969), 71–93. Google Scholar

[32] [32] Pedersen, G. K., C*-Algebras and their automorphism groups. London Mathematical Society Monographs, 14, Academic Press, London-New York, 1979. Google Scholar

[33] [33] Pinzari, C., Watatani, Y., and Yonetani, K., KMS states, entropy and the variational principle in full C*-dynamical systems. Comm. Math. Phys. 213(2000), no. 2, 331–379. Google Scholar | DOI

[34] [34] Popa, S., Strong rigidity of II1 factors arising from malleable actions of w-rigid groups. I. Invent. Math. 165(2006), no. 2, 369–408. Google Scholar | DOI

[35] [35] Popa, S. and Vaes, S., Actions of F1 whose II1 factors and orbit equivalence relations have prescribed fundamental group. J. Amer. Math. Soc. 23(2010), no. 2, 383–403. Google Scholar | DOI

[36] [36] Řadulescu, F., The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R_ +. J. Amer. Math. Soc. 5(1992), no. 3, 517–532. Google Scholar

[37] [37] Razak, S., On the classification of simple stably projectionless C*-algebras. Canad. J. Math. 54(2002), no. 1, 138–224. Google Scholar | DOI

[38] [38] Raeburn, I. and D. P.Williams, Morita equivalence and continuous-trace C*-algebras. Mathematical Surveys and Monographs, 60, American Mathematical Society, Providence, RI, 1998. Google Scholar

[39] [39] Rieffel, M. A., Morita equivalence for operator algebras. In: Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math., 38, American Mathematical Society, Providence, RI, 1982, pp. 285–298. Google Scholar

[40] [40] Tsang, K.-W., On the positive tracial cones of simple stably projectionless C*-algebras. J. Funct. Anal. 227(2005), no. 1, 188–199. Google Scholar | DOI

[41] [41] Voiculescu, D., Circular and semicircular systems and free product factors. In: Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989), Progr. Math. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 45–60. Google Scholar

[42] [42] Watatani, Y., Index for C*-subalgebras. Mem. Amer. Math. Soc. 83(1990), no. 424. Google Scholar