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Properly infinite \(C(X)\)-algebras and \(K_{1}\)-injectivity. (English) Zbl 1157.46032

Summary: We investigate if a unital \(C(X)\)-algebra is properly infinite when all its fibres are properly infinite. We show that this question can be rephrased in several different ways, including the question of whether every unital properly infinite \(C^{*}\)-algebra is \(K_{1}\)-injective. We provide partial answers to these questions, and we show that the general question on proper infiniteness of \(C(X)\)-algebras can be reduced to establishing proper infiniteness of a specific \(C([0,1])\)-algebra with properly infinite fibres.

MSC:

46L35 Classifications of \(C^*\)-algebras
46L05 General theory of \(C^*\)-algebras
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[1] B. Blackadar and D. Handelman, Dimension functions and traces on C -algebras. J. Funct. Anal. 45 (1982), 297-340. · Zbl 0513.46047 · doi:10.1016/0022-1236(82)90009-X
[2] E. Blanchard, A few remarks on exact C.X /-algebras. Rev. Roumaine Math. Pures Appl. 45 (2000), 565-576 (2001). · Zbl 0994.46016
[3] E. Blanchard and E. Kirchberg, Global Glimm halving for C -bundles. J. Operator Theory 52 (2004), 385-420. · Zbl 1073.46509
[4] J. Cuntz, K-theory for certain C -algebras. Ann. of Math. (2) 113 (1981), 181-197. · Zbl 0437.46060 · doi:10.2307/1971137
[5] M. Dadarlat, Continuous fields of C -algebras over finite dimensional spaces. Preprint. · Zbl 1190.46040 · doi:10.1016/j.aim.2009.06.019
[6] J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C -algèbres. Bull. Soc. Math. France 91 (1963), 227-284. · Zbl 0127.33102
[7] D. Handelman, Homomorphisms of C algebras to finite AW algebras. Michigan Math. J. 28 (1981), 229-240. · Zbl 0468.46045 · doi:10.1307/mmj/1029002512
[8] I. Hirshberg, M. Rørdam, and W. Winter, C0.X/-algebras, stability and strongly self- absorbing C -algebras. Math. Ann. 339 (2007), 695-732. · Zbl 1128.46020 · doi:10.1007/s00208-007-0129-8
[9] M. A. Rieffel, The homotopy groups of the unitary groups of non-commutative tori. J. Operator Theory 17 (1987), 237-254. · Zbl 0656.46056
[10] M. Rørdam, Classification of inductive limits of Cuntz algebras. J. Reine Angew. Math. 440 (1993), 175-200. · Zbl 0783.46031 · doi:10.1515/crll.1993.440.175
[11] M. Rørdam, On sums of finite projections. In Operator algebras and operator theory (Shanghai, China, 1997), Contemp. Math. 228, Amer. Math. Soc., Providence, RI, 1998, 327-340. · Zbl 0927.46030
[12] M. Rørdam, A simple C -algebra with a finite and an infinite projection. Acta Math. 191 (2003), 109-142. · Zbl 1072.46036 · doi:10.1007/BF02392697
[13] M. Rørdam, F. Larsen, and N. Laustsen, An introduction to K -theory for C -algebras . London Math. Soc. Stud. Texts 49, Cambridge University Press, Cambridge 2000. · Zbl 0967.19001
[14] A. S. Toms and W. Winter, Strongly self-absorbing C -algebras. Trans. Amer. Math. Soc. 359 (2007), 3999-4029. · Zbl 1120.46046 · doi:10.1090/S0002-9947-07-04173-6
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