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Open projections and suprema in the Cuntz semigroup

Published online by Cambridge University Press:  27 September 2016

JOAN BOSA ,
GABRIELE TORNETTA  and
JOACHIM ZACHARIAS
JOAN BOSA
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, G12 8QW, Glasgow e-mails: [email protected], [email protected], [email protected]
GABRIELE TORNETTA
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, G12 8QW, Glasgow e-mails: [email protected], [email protected], [email protected]
JOACHIM ZACHARIAS
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, G12 8QW, Glasgow e-mails: [email protected], [email protected], [email protected]
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Abstract

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated in [12]. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary C*-subalgebra of A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 1 , January 2018 , pp. 135 - 146
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Akemann, C. A. The general Stone-Weierstrass problem. J. Funct. Anal. 4 (2) (1969), 277–294.CrossRefGoogle Scholar
[2] Antoine, R., Bosa, J. and Perera, F. Completions of monoids with applications to the Cuntz semigroup. Internat. J. Math. 22 (6) (2011), 837–861.CrossRefGoogle Scholar
[3] Antoine, R., Perera, F. and Thiel, H. Tensor products and regularity properties of Cuntz semigroups. Preprint.Google Scholar
[4] Ara, P., Perera, F. and Toms, A. S. K-theory for operator algebras. Classification of C*-algebras. In Aspects of operator algebras and applications. Contemp. Math. vol. 534 (Amer. Math. Soc., Providence, RI, 2011), pp. 1–71.CrossRefGoogle Scholar
[5] Blackadar, B. K-theory for operator algebras. Math. Sci. Res. Inst. Publ. vol. 5 (Cambridge University Press, Cambridge, second edition, 1998).Google Scholar
[6] Brown, N. P. and Ciuperca, A. Isomorphism of Hilbert modules over stably finite C*-algebras. J. Funct. Anal. 257 (1) (2009), 332–339.CrossRefGoogle Scholar
[7] Brown, N. P., Perera, F. and Toms, A. S. The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras. J. Reine Angew. Math 621 (2008), 191–211.Google Scholar
[8] Coward, K. T., Elliott, G. A. and Ivanescu, C. The Cuntz semigroup as an invariant for C*-algebras. J. Reine Angew. Math. 623 (2008), 161–193.Google Scholar
[9] Cuntz, J. Dimension functions on simple C*-algebras. Math. Ann. 233 (1978), 145–153.CrossRefGoogle Scholar
[10] Cuntz, J. The internal structure of simple C*-algebras. In Operator algebras and applications, Part I (Kingston, Ont., 1980), Proc. Sympos. Pure Math. vol. 38 (Amer. Math. Soc., Providence, R.I., 1982), pp. 85–115.CrossRefGoogle Scholar
[11] Kirchberg, E. and Rørdam, M.. Infinite Non-simple C*-Algebras: Absorbing the Cuntz Algebra $\Ncal O_\infty$ . Adv. Math. 167 (2) (2002), 195–264.CrossRefGoogle Scholar
[12] Ortega, E., Rørdam, M. and Thiel, H.. The Cuntz semigroup and comparison of open projections. J. Funct. Anal. 260 (12) (2011), 3474–3493.CrossRefGoogle Scholar
[13] Peligrad, C. and Zsidó, L. Open projection of C*-algebras comparison and regularity. In Operator theoretical methods. Proceedings of the 17th international conference on operator theory (Timişoara, Romania, June 23–26, 1998), (Bucharest, 2000. The Theta Foundation), pp. 285–300.Google Scholar
[14] Robert, L. and Tikuisis, A. Hilbert C*-modules over a commutative C*-algebra. Proc. Lond. Math. Soc. (3) 102 (2) (2011), 229–256.CrossRefGoogle Scholar
[15] Rørdam, M. On the structure of simple C*-algebras tensored with a UHF-algebra. II. J. Funct. Anal. 107 (2) (1992), 255–269.CrossRefGoogle Scholar
[16] Santiago, L. Classification of non-simple C*-algebras: Inductive limits of splitting interval algebras. PhD thesis (Department of Mathematics, University of Toronto, Toronto, Canada, 2008).Google Scholar