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EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  08 June 2009

SHUDONG LIU  and
XIAOCHUN FANG
SHUDONG LIU*
Affiliation:
School of Mathematical Sciences, Qufu Normal University, Qufu, People’s Republic of China (email: [email protected])
XIAOCHUN FANG
Affiliation:
Department of Mathematics, Tongji University, Shanghai, People’s Republic of China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

Keywords

C*-algebraextensionCuntz algebra

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 1 , August 2009 , pp. 83 - 90
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was supported by the National Natural Science Foundation of China (grant no. 10771161) and the Natural Science Foundation of Shandong Province (grant no. Y2006A03).

References

[1] Blackadar, B., K-Theory for Operator Algebras (Springer, New York, 1986).CrossRefGoogle Scholar
[2] Brown, L. G. and Dadarlat, M., ‘Extensions of C *-algebras and quasidiagonality’, J. London Math. Soc. (2) 53 (1996), 582–600.CrossRefGoogle Scholar
[3] Brown, L. G., Douglas, R. G. and Fillmore, P. A., ‘Unitary equivalence modulo the compact operators and extensions of C *-algebras’, in: Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, 1973), Lecture Notes in Mathematics, 345 (Springer, Berlin, 1973), pp. 58–128.CrossRefGoogle Scholar
[4] Brown, L. G., Douglas, R. G. and Fillmore, P. A., ‘Extensions of C *-algebras and K-homology’, Ann. of Math. (2) 105(2) (1977), 265–324.CrossRefGoogle Scholar
[5] Busby, R. C., ‘Double centralizers and extensions of C *-algebras’, Trans. Amer. Math. Soc. 132 (1968), 79–99.Google Scholar
[6] Cuntz, J., ‘Simple C *-algebras generated by isometries’, Comm. Math. Phys. 52(2) (1977), 173–185.CrossRefGoogle Scholar
[7] Cuntz, J., ‘K-theory for certain C *-algebras’, Ann. of Math. (2) 113 (1981), 181–197.CrossRefGoogle Scholar
[8] Lin, H., ‘On the classification of C *-algebras of real rank zero with zero K 1’, J. Operator Theory 35 (1996), 147–178.Google Scholar
[9] Lin, H., An Introduction to the Classification of Amenable C *-algebras (World Scientific, Singapore, 2001).Google Scholar
[10] Lin, H., ‘Full extension and approximate unitary equivalences’, Pacific J. Math. 229(2) (2007), 389–428.CrossRefGoogle Scholar
[11] Lin, H. and Phillips, N. C., ‘Approximate unitary equivalence of homomorphisms from Ø’, J. Reine Angew. Math. 464 (1995), 173–186.Google Scholar
[12] Liu, S. and Fang, X., ‘Extension algebras of Cuntz algebra’, J. Math. Anal. Appl. 329 (2007), 655–663.CrossRefGoogle Scholar
[13] Liu, S. and Fang, X., ‘K-theory for extensions of purely infinite simple C *-algebras’, Chinese Ann. of Math. Ser. A 29(2) (2008), 195–202.Google Scholar
[14] Liu, S. and Fang, X., ‘K-theory for extensions of purely infinite simple C *-algebras: II’, Chinese Ann. of Math. Ser. A to appear.Google Scholar
[15] Paschke, W. L. and Salinas, N., ‘Matrix algebras over 𝒪n’, Michigan Math. J. 26(1) (1979), 3–12.CrossRefGoogle Scholar