On the Lack of Inverses to C*-Extensions Related to Property T Groups

Manuilov, V. ; Thomsen, K.

Canadian mathematical bulletin, Tome 50 (2007) no. 2, pp. 268-283

Résumé

Using ideas of S. Wassermann on non-exact ${{C}^{*}}$ -algebras and property $\text{T}$ groups, we show that one of his examples of non-invertible ${{C}^{*}}$ -extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a ${{C}^{*}}$ -extension which is not even invertible up to homotopy.

Détail

DOI : 10.4153/CMB-2007-027-1

Mots-clés : 19K33, 46L06, 46L80, 20F99, C*-algebra extension, property T group, asymptotic tensor, C*-norm, homotopy

Bibliographie

[1] [1] Anderson, J., A C*-algebra for which Ext() is not a group. Ann. of Math. 107(1978), no. 3, 455–458. Google Scholar

[2] [2] Brown, L. G., Douglas, R. G., and Fillmore, P. A., Extensions of C*-algebras and K-homology. Ann. of Math. 105(1977), no. 2, 265–324. Google Scholar

[3] [3] Connes, A. and Higson, N., Déformations, morphismes asymptotiques et K-théorie bivariante. C. R. Acad. Sci. Paris Sér. I Math. 311(1990), no. 2, 101–106. Google Scholar

[4] [4] Dadarlat, M. and Eilers, S., On the classification of nuclear C*-algebras. Proc. London Math. Soc. (3) 85(2002), no. 1, 168–210. Google Scholar

[5] [5] Effros, E. G., Dimensions and C*-Algebras, CBMS Regional Conference Series 46, American Mathematical Society, Providence, RI, 1981. Google Scholar

[6] [6] Haagerup, U. and Thorbjornsen, S., A new application of random matrices: Ext is not a group. Ann. of Math. 162(2005), no. 2, 711–775. Google Scholar

[7] [7] de la Harpe, P., Robertson, A. G., and Valette, A., On the spectrum of the sum of generators for a finitely generated group. Israel J. Math. 81(1993), no. 1–2, 65–96. Google Scholar

[8] [8] de la Harpe, P. and Valette, A., La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque, . Google Scholar

[9] [9] Kirchberg, E., On non semi-split extensions, tensor products and exactness of C*-algebras. Invent. Math. 112(1993), 449–489. Google Scholar

[10] [10] Manuilov, V. and Thomsen, K., Asymptotically split extensions and E-theory. Algebra i Analiz (2000) no. 5, 142–157 (in Russian); English translation: St. Petersburg Math. J. 12(2001), no. 5, 819–830. Google Scholar

[11] [11] Manuilov, V. and Thomsen, K., The Connes–Higson construction is an isomorphism. J. Funct. Anal. 213(2004), no. 1, 154–175. Google Scholar

[12] [12] Manuilov, V. and Thomsen, K., On the asymptotic tensor C*-norm. Arch. Math. 86(2006), no. 2, 138–144. Google Scholar

[13] [13] Ozawa, N., An application of expanders to (l) ⊗ (l). J. Funct. Anal. 198(2003), no. 2, 499–510. Google Scholar

[14] [14] Valette, A., Minimal projections, integrable representations and property T . Arch. Math. 43(1984), no. 5, 397–406. Google Scholar

[15] [15] Voiculescu, D., A non-commutative Weyl–von Neumann theorem. Rev. Roumaine Math. Pures. Appl. 21(1976), no. 1, 97–113. Google Scholar

[16] [16] Wang, P. S., On isolated points in the dual space of locally compact groups. Math. Ann. 218(1975), no. 1, 19–34. Google Scholar

[17] [17] Wassermann, S., Liftings in C*-algebras: a counterexample. Bull. London Math. Soc. 9(1976), no. 2, 201–202. Google Scholar

[18] [18] Wassermann, S., Tensor products of free group C*-algebras. Bull. London Math. Soc. 22(1990), no. 4, 375–380. Google Scholar

[19] [19] Wassermann, S., C*-algebras associated with groups with Kazhdan's property T.. Ann. of Math. 134(1991), no. 2, 423–431. Google Scholar

[20] [20] Wassermann, S., Exact C*-Algebras and Related Topics. Lecture Notes Series 19, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1994 Google Scholar