Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T15:52:40.090Z Has data issue: false hasContentIssue false

VOICULESCU’S THEOREM FOR NONSEPARABLE $\text{C}^{\ast} $-ALGEBRAS

Published online by Cambridge University Press:  20 July 2020

ANDREA VACCARO*
Affiliation:
DEPARTMENT OF MATHEMATICS BEN-GURION UNIVERSITY OF THE NEGEV P.O.B. 653, BE’ER SHEVA84105, ISRAELE-mail: [email protected]: http://www.math.wisc.edu/~lempp/
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that Voiculescu’s noncommutative version of the Weyl-von Neumann Theorem can be extended to all unital, separably representable $\mathrm {C}^\ast $ -algebras whose density character is strictly smaller than the (uncountable) cardinal invariant $\mathfrak {p}$ . We show moreover that Voiculescu’s Theorem consistently fails for $\mathrm {C}^\ast $ -algebras of larger density character.

Type
Articles
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

REFERENCES

Arveson, W., Notes on extensions of ${C}^{\ast }$ -algebras . Duke Mathematical Journal, vol. 44 (1977), no. 2, pp. 329355.CrossRefGoogle Scholar
Bartoszyński, T. and Judah, H., Set Theory, A. K. Peters, Ltd., Wellesley, MA, 1995, On the structure of the real line.CrossRefGoogle Scholar
Bell, M. G., On the combinatorial principle $P \mathfrak{c}$ . Fundamenta Mathematicae, vol. 114 (1981), no. 2, pp. 149157.CrossRefGoogle Scholar
Berger, C. A. and Coburn, L. A., On Voiculescu’s double commutant Theorem . Proceedings of the American Mathematical Society, vol. 124 (1996), no. 11, pp. 34533457.CrossRefGoogle Scholar
Blackadar, B., $\textbf{K-}\!$ Theory for Operator Algebras, vol. 3, second ed., Mathematical Sciences Research Institute Publications, Cambridge University Press, Cambridge, 1998.Google Scholar
Brown, L. G., Douglas, R. G., and Fillmore, P. A., Extensions of ${C}^{\ast }$ -algebras and $\; K$ -homology . Annals of Mathematics, vol. 105 (1977), no. 2, pp. 265324.CrossRefGoogle Scholar
Brown, N. P. and Ozawa, N., $\textbf{C}^{\ast}\!\!$ -algebras and finite-dimensional approximations , Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008.CrossRefGoogle Scholar
Choi, M. D. and Effros, E. G., The completely positive lifting problem for ${C}^{\ast }$ -algebras . Annals of Mathematics, vol. 104 (1976), no. 3, pp. 585609.CrossRefGoogle Scholar
Davidson, K. R., Essentially normal operators , A Glimpse at Hilbert Space Operators: Paul R. Halmos in Memoriam, Operator Theory: Advances and Applications, vol. 207, Birkhäuser Verlag, Basel, 2010, pp. 209222.CrossRefGoogle Scholar
Farah, I., A new bicommutant Theorem . Pacific Journal of Mathematics, vol. 288 (2017), no. 1, pp. 6985.CrossRefGoogle Scholar
Farah, I., Hirshberg, I., and Vignati, A., The Calkin algebra is ${\aleph}_1$ -universal, preprint, 2018, arXiv:1707.01782 (to appear in Israel J. Math.).Google Scholar
Farah, I., Katsimpas, G., and Vaccaro, A., Embedding $C^{\ast}$ -algebras into the Calkin algebra, preprint, 2018, arXiv:1810.00255 (to appear in Int. Math. Res. Not.).Google Scholar
Farah, I. and Wofsey, E., Set theory and operator algebras . London Mathematical Society Lecture Note Series, vol. 406 (2012), pp. 63120.Google Scholar
Hadwin, D. W., Nonseparable approximate equivalence . Transactions of the American Mathematical Society, vol. 266 (1981), no. 1, pp. 203231.CrossRefGoogle Scholar
Higson, N. and Roe, J., Analytic $K\!\!\!\!$ -Homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.Google Scholar
von Neumann, J., Charakterisierung des Spektrums eines Integraloperators. Revue de Métaphysique et de Morale, vol. 42 (1935), no. 4, pp. 1719.Google Scholar
Voiculescu, D., A non-commutative Weyl-von Neumann Theorem . Revue Roumaine de Mathématique Pures et Appliquées, vol. 21 (1976), no. 1, pp. 97113.Google Scholar
Weyl, H., Über beschränkte quadratische Formen, deren Differenz vollstetig ist . Rendiconti del Circolo Matematico di Palermo, vol. 27 (1909), no. 1, pp. 373392.CrossRefGoogle Scholar