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THE UNITAL EXT-GROUPS AND CLASSIFICATION OF C*-ALGEBRAS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  13 March 2019

JAMES GABE Open the ORCID record for JAMES GABE [Opens in a new window]  and
EFREN RUIZ
JAMES GABE*
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8SQ, Scotland e-mail: jamiegabe123@hotmail.com
EFREN RUIZ
Affiliation:
Department of Mathematics, University of Hawaii, Hilo, 200 W. Kawili St., Hilo, Hawaii, 96720-4091USA e-mail: ruize@hawaii.edu
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Abstract

The semigroups of unital extensions of separable C*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K-theoretic classification of both unital and non-unital extensions of C*-algebras, and in particular we obtain a complete K-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

MSC classification

Primary: 46L35: Classifications of $C^*$-algebras
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 201 - 231
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

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References

Blackadar, B., K-theory for operator algebras, vol. 5 of Mathematical Sciences Research Institute Publications, 2nd edition, (Cambridge University Press, Cambridge, 1998).Google Scholar
Blanchard, E., Rohde, R. and Rørdam, M., Properly infinite C(X) -algebras and K 1-injectivity, J. Noncommut. Geom. 2(3) (2008), 263–282.CrossRefGoogle Scholar
Brown, L. G., Semicontinuity and multipliers of C*-algebras, Can. J. Math. 40(4) (1988), 865–988.Google Scholar
Brown, L. G. and Pedersen, G. K., C*-algebras of real rank zero, J. Funct. Anal. 99(1) (1991), 131–149.Google Scholar
Choi, M. D. and Effros, E. G., The completely positive lifting problem for C*-algebras, Ann. Math. 104(3) (1976), 585–609.CrossRefGoogle Scholar
Cuntz, J., K-theory for certain C*-algebras, Ann. Math. 113(1)(1981) 181–197.CrossRefGoogle Scholar
Cuntz, J. and Higson, N., Kuiper’s theorem for Hilbert modules. in Operator algebras and mathematical physics (Iowa City, Iowa, 1985), vol. 62 of Contemp. Math. (Amer. Math. Soc., Providence, RI, 1987), 429–435.CrossRefGoogle Scholar
Effros, E. G., Dimensions and C*-algebras, vol. 46 of CBMS Regional Conference Series in Mathematics, (Conference Board of the Mathematical Sciences, Washington, D.C., 1981).CrossRefGoogle Scholar
Eilers, S., Gabe, J., Katsura, T., Ruiz, E. and Tomforde, M., The extension problem for graph C*-algebras, (2018). arxiv.1810.12147 .Google Scholar
Eilers, S., Loring, T. A. and Pedersen, G. K., Morphisms of extensions of C*-algebras: pushing forward the Busby invariant, Adv. Math. 147(1)(1999), 74–109.CrossRefGoogle Scholar
Eilers, S., Restorff, G. and Ruiz, E., Classification of extensions of classifiable C*-algebras, Adv. Math. 222(6) (2009), 2153–2172.CrossRefGoogle Scholar
Eilers, S., Restorff, G., Ruiz, E. and Sørensen, A., The complete classification of unital graph C*-algebras: geometric and strong, (2016). Preprint, arxiv.1611.07120.Google Scholar
Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra. 38(1) (1976), 29–44.CrossRefGoogle Scholar
Elliott, G. A., Gong, G., Lin, H. and Niu, Z., On the classification of simple C*-algebras with finite decomposition rank, II. (2015). arxiv.1507.03437v2 .Google Scholar
Elliott, G. A. and Kucerovsky, D., An abstract Voiculescu–Brown–Douglas–Fillmore absorption theorem, Pacific J. Math. 198(2) (2001), 385–409.CrossRefGoogle Scholar
Gabe, J., A note on nonunital absorbing extensions, Pac. J. Math. 284(2) (2016), 383–393.CrossRefGoogle Scholar
Gong, G., Lin, H. and Niu, Z., Classification of finite simple amenable ${\cal Z}$-stable C*-algebras, (2015). arxiv.1501.00135v4.Google Scholar
Kasparov, G. G., The operator K-functor and extensions of C*-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44(3) (1980) 571–636, 719.Google Scholar
Kirchberg, E., The classification of purely infinite C*-algebras using Kasparov’s theory, 1994.Google Scholar
Kucerovsky, D. and Ng, P. W., The corona factorization property and approximate unitary equivalence, Houston J. Math. 32(2) (2006) 531–550(electronic).Google Scholar
Manuilov, V. and Thomsen, K., The group of unital C*-extensions, in C*-algebras and elliptic theory, (Trends Math.) (Birkhäuser, Basel, 2006), 151–156.CrossRefGoogle Scholar
Nagy, G., Some remarks on lifting invertible elements from quotient C*-algebras, J. Operat. Theory. 21(2) (1989), 379–386.Google Scholar
Nistor, V., Stable range for tensor products of extensions of ${\cal K}$ by C(X), J. Operat. Theory. 16(2) (1986), 387–396.Google Scholar
Ortega, E., Perera, F. and Rørdam, M., The corona factorization property, stability, and the Cuntz semigroup of a C*-algebra, Int. Math. Res. Not. IMRN. (1) (2012), 34–66.CrossRefGoogle Scholar
Perera, F., Toms, A., White, S. and Winter, W., The Cuntz semigroup and stability of close C*-algebras, Anal. PDE, 7(4) (2014), 929–952.CrossRefGoogle Scholar
Phillips, N. C., A classification theorem for nuclear purely infinite simple C*-algebras, Doc. Math. 5 (2000), 49–114 (electronic).Google Scholar
Rieffel, M. A., Dimension and stable rank in the K-theory of C*-algebras, Proc. London Math. Soc. (3) 46(2) (1983), 301–333.CrossRefGoogle Scholar
Robert, L., Nuclear dimension and n-comparison, Münster J. Math. 4(2011), 65–71.Google Scholar
Rørdam, M., Classification of extensions of certain C*-algebras by their six term exact sequences in K-theory, Math. Ann. 308(1) (1997), 93–117.Google Scholar
Rørdam, M., Larsen, F. and Laustsen, N., An introduction to K-theory for C*-algebras, vol. 49 of London Mathematical Society Student Texts, (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar
Rosenberg, J. and Schochet, C., The Künneth theorem and the universal coefficient theorem for Kasparov’s generalized K-functor, Duke Math. J. 55(2) (1987), 431–474.CrossRefGoogle Scholar
Skandalis, G., On the strong Ext bifunctor, (1984), preprint.Google Scholar
Skandalis, G., Une notion de nucléarité en K-théorie (d’après J. Cuntz), K-Theory. 1(6) (1988), 549–573.CrossRefGoogle Scholar
Thomsen, K., On absorbing extensions, Proc. Am. Math. Soc. 129(5) (2001), 1409–1417(electronic).CrossRefGoogle Scholar
Tikuisis, A., White, S. and Winter, W., Quasidiagonality of nuclear C*-algebras, Ann. Math. (2) 185(1) (2017) 229–284.CrossRefGoogle Scholar
Wei, C., On the classification of certain unital extensions of C*-algebras, Houston J. Math. 41(3)(2015), 965–991.Google Scholar