Chern Characters of Fourier Modules

Walters, Samuel G.

Canadian journal of mathematics, Tome 52 (2000) no. 3, pp. 633-672

Résumé

Let ${{A}_{\theta}}$ denote the rotation algebra—the universal ${{C}^{*}}$ -algebra generated by unitaries $U,V$ satisfying $VU={{e}^{2\pi i\theta }}UV$ , where $\theta $ is a fixed real number. Let $\sigma $ denote the Fourier automorphism of ${{A}_{\theta}}$ defined by $U\mapsto V\text{,}V\mapsto {{U}^{-1}}$ , and let ${{B}_{\theta }}={{A}_{\theta }}{{\rtimes }_{\sigma }}\mathbb{Z}/4\mathbb{Z}$ denote the associated ${{C}^{*}}$ -crossed product. It is shown that there is a canonical inclusion ${{\mathbb{Z}}^{9}}\hookrightarrow {{K}_{0}}({{B}_{\theta }})$ for each $\theta $ given by nine canonical modules. The unbounded trace functionals of ${{B}_{\theta }}$ (yielding the Chern characters here) are calculated to obtain the cyclic cohomology group of order zero $\text{H}{{\text{C}}^{0}}({{B}_{\theta }})$ when $\theta $ is irrational. The Chern characters of the nine modules—and more importantly, the Fourier module—are computed and shown to involve techniques from the theory of Jacobi’s theta functions. Also derived are explicit equations connecting unbounded traces across strong Morita equivalence, which turn out to be non-commutative extensions of certain theta function equations. These results provide the basis for showing that for a dense ${{\text{G}}_{\delta }}$ set of values of $\theta $ one has ${{K}_{0}}({{B}_{\theta }})\cong {{\mathbb{Z}}^{9}}$ and is generated by the nine classes constructed here.

Détail

DOI : 10.4153/CJM-2000-028-9

Mots-clés : 46L80, 46L40, C *-algebras, unbounded traces, Chern characters, irrational rotation algebras, K-groups

Bibliographie

[1] [1] Boca, F. P., On the flip fixed point algebra in certain noncommutative tori. Indiana J. Math. 45(1996), 253–272. Google Scholar

[2] [2] Boca, F. P., Projections in rotation algebras and theta functions. Comm. Math. Phys. (1999), to appear. Google Scholar

[3] [3] Bratteli, O., Elliott, G. A., Evans, D. E. and Kishimoto, A., Non-commutative spheres I. Internat. J. Math. (2) 2(1990), 139–166. Google Scholar

[4] [4] Bratteli, O., Elliott, G. A., Evans, D. E. and Kishimoto, A., Non-commutative spheres II: rational rotations. J. Operator Theory 27(1992), 53–85. Google Scholar

[5] [5] Bratteli, O. and Kishimoto, A., Non-commutative spheres III. Irrational Rotations. Comm. Math. Phys. 147(1992), 605–624. Google Scholar

[6] [6] Connes, A., C*algebre et geometrie differentielle . C. R. Acad. Sci. Paris Ser. A–B 290(1980), 599–604. Google Scholar

[7] [7] Connes, A., Noncommutative Geometry . Academic Press, 1994. Google Scholar

[8] [8] Elliott, G. and Evans, D., The structure of the irrational rotation C*-algebra. Ann. of Math. 138(1993), 477–501. Google Scholar

[9] [9] Farsi, C. and Watling, N., Quartic algebras. Canad. J. Math. (6) 44(1992), 1167–1191. Google Scholar

[10] [10] Jeong, J. A. and Osaka, H., Extremally rich C*-crossed products and cancellation property. J. Austral. Math. Soc. Ser. A (3) 64(1998), 285–301. Google Scholar

[11] [11] Rieffel, M., C*-algebras associated with irrational rotations. Pacific J. Math. (2) 93(1981), 415–429. Google Scholar

[12] [12] Rieffel, M., The cancellation theoremfor projective modules over irrational rotation algebras. Proc. LondonMath. Soc. (3) 47(1983), 285–302. Google Scholar

[13] [13] Rudin, W., Functional Analysis. McGraw-Hill, 2nd edition, 1991. Google Scholar

[14] [14] Walters, S., Projective modules over the non-commutative sphere. J. LondonMath. Soc. (2) 51(1995), 589–602. Google Scholar

[15] [15] Walters, S., Inductive limit automorphisms of the irrational rotation algebra. Comm. Math. Phys. 171(1995), 365–381. Google Scholar

[16] [16] Walters, S., K-theory of non commutative spheres arising from the Fourier automorphism. (1999), 39 pages, preprint. Google Scholar

[17] [17] Walters, S., On the irrational quartic algebra. C. R. Math. Rep. Acad. Sci. Canada (1999), 5 pages, to appear. Google Scholar

[18] [18] Whittaker, E. T. and Watson, G. N., A course in modern analysis. Cambridge Univ. Press, 4th edition, 1950. Google Scholar