Symmetries of Kirchberg Algebras

Benson, David J. ; Kumjian, Alex ; Phillips, N. Christopher

Canadian mathematical bulletin, Tome 46 (2003) no. 4, pp. 509-528

Résumé

Let ${{G}_{0}}$ and ${{G}_{1}}$ be countable abelian groups. Let ${{\gamma }_{i}}$ be an automorphism of ${{G}_{i}}$ of order two. Then there exists a unital Kirchberg algebra $A$ satisfying the Universal Coefficient Theorem and with $[{{1}_{A}}]\,=\,0$ in ${{K}_{0}}(A)$ , and an automorphism $\alpha \,\in \,\text{Aut}(A)$ of order two, such that ${{K}_{0}}(A)\,\cong \,{{G}_{0}}$ , such that ${{K}_{1}}(A)\,\cong \,{{G}_{1}}$ , and such that ${{\alpha }_{*}}\,:\,{{K}_{i}}(A)\,\to \,{{K}_{i}}(A)$ is ${{\gamma }_{i}}$ . As a consequence, we prove that every ${{\mathbb{Z}}_{2}}$ -graded countable module over the representation ring $R({{\mathbb{Z}}_{2}})$ of ${{\mathbb{Z}}_{2}}$ is isomorphic to the equivariant $K$ -theory ${{K}^{{{\mathbb{Z}}_{2}}}}(A)$ for some action of ${{\mathbb{Z}}_{2}}$ on a unital Kirchberg algebra $A$ .Along the way, we prove that every not necessarily finitely generated $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$ -module which is free as a $\mathbb{Z}$ -module has a direct sum decomposition with only three kinds of summands, namely $\mathbb{Z}\left[ {{\mathbb{Z}}_{2}} \right]$ itself and $\mathbb{Z}$ on which the nontrivial element of ${{\mathbb{Z}}_{2}}$ acts either trivially or by multiplication by −1.

Détail

DOI : 10.4153/CMB-2003-049-7

Mots-clés : 20C10, 46L55, 19K99, 19L47, 46L40, 46L80

Bibliographie

[1] [1] Bass, H., Big projective modules are free. Illinois J. Math. 7 (1963), 24–31. Google Scholar

[2] [2] Benson, D. J. and Goodearl, K. R., Periodic flat modules, and flat modules for finite groups. Pacific J. Math. 196 (2000), 45–67. Google Scholar

[3] [3] Blackadar, B., K-Theory for Operator Algebras. MSRI Publication Series 5, Springer-Verlag, New York-Heidelberg-Berlin-Tokyo, 1986. Google Scholar

[4] [4] Butler, M. C. R., Campbell, J. M. and Kovács, L. G., Infinite rank integral representations of groups and orders of finite lattice type. In preparation. Google Scholar

[5] [5] Butler, M. C. R. and Kovács, L. G., Large lattices over classical orders of finite lattice type. Draft preprint. Google Scholar

[6] [6] Cuntz, J., Simple C*-algebras generated by isometries. Comm. Math. Phys. 57 (1977), 173–185. Google Scholar

[7] [7] Cuntz, J., K-theory for certain C*-algebras. Ann. Math. 113 (1981), 181–197. Google Scholar

[8] [8] Curtis, C. W. and Reiner, I., Representation Theory of Finite Groups and Associative Algebras. Interscience Publishers, New York-London-Sydney, 1962. Google Scholar

[9] [9] Elliott, G. A. and Rørdam, M., Classification of certain infinite simple C*-algebras II. Comment. Math. Helv. 70 (1995), 615–638. Google Scholar

[10] [10] Fuchs, L., Infinite Abelian Groups, Vol. I. Academic Press, New York-London, 1970. Google Scholar

[11] [11] Hua, L. K. and Reiner, I., Automorphisms of the unimodular group. Trans. Amer.Math. Soc. 71 (1951), 331–348. Google Scholar

[12] [12] Jeong, J. A. and Osaka, H., Extremally rich C*-crossed products and the cancellation property. J. Austral. Math. Soc. (Ser. A) 64 (1998), 285–301. Google Scholar

[13] [13] Kirchberg, E., Commutants of unitaries in UHF algebras and functorial properties of exactness. J. Reine Angew.Math. 452 (1994), 39–77. Google Scholar

[14] [14] Kirchberg, E., The classification of purely infinite C*-algebras using Kasparov's theory. Preliminary preprint (3rd draft). Google Scholar

[15] [15] Kumjian, A., On certain Cuntz-Pimsner algebras. Preprint. Google Scholar

[16] [16] Mac Lane, S., Categories for the Working Mathematician. Graduate Texts in Math. 5, Springer-Verlag, New York-Heidelberg-Berlin, 1971. Google Scholar

[17] [17] Pedersen, G. K.,C*-Algebras and their Automorphism Groups. Academic Press, London-New York-San Francisco, 1979. Google Scholar

[18] [18] Phillips, N. C., Equivariant K-Theory and Freeness of Group Actions on C*-Algebras. Lecture Notes in Math. 1274, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo, 1987. Google Scholar

[19] [19] Phillips, N. C., A classification theorem for nuclear purely infinite simple C*-algebras. Doc. Math. 5 (2000), 49–114 (electronic). Google Scholar

[20] [20] Pimsner, M., A class of C*-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z. In: Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Commun. 12, Amer.Math. Soc., Providence, RI, 1997, 189–212. Google Scholar

[21] [21] Rieffel, M. A., Actions of finite groups on C*-algebras. Math. Scand. 47 (1980), 157–176. Google Scholar

[22] [22] Rørdam, M. and Størmer, E., Classification of nuclear C*-algebras. Entropy in operator algebras. Springer-Verlag, Berlin, 2002. Google Scholar

[23] [23] Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov's generalized K-functor. Duke Math. J. 55 (1987), 431–474. Google Scholar

[24] [24] Schochet, C., Topological methods for C*-algebras II: geometric resolutions and the Künneth formula. Pacific J. Math. 98 (1982), 443–458. Google Scholar

[25] [25] Schochet, C., Topological methods for C*-algebras III: axiomatic homology. Pacific J. Math. 114 (1984), 399–445. Google Scholar

[26] [26] Swan, R. G., Induced representations and projective modules. Ann. of Math. 71 (1960), 552–578. Google Scholar

[27] [27] Takai, H., On a duality for crossed products of C*-algebras. J. Funct. Anal. 19 (1975), 25–39. Google Scholar