Abstract
Let \( 0 \xrightarrow {}J \xrightarrow { } A \xrightarrow { }B \xrightarrow {} 0\) be an extension of \(C^*\)-algebras. Suppose that both J and B have tracial rank no more than one. It is shown that A has tracial topological rank no more than one whenever it is a quasidiagonal extension, and A has property \((P_1)\) if the extension is tracially quasidiagonal.
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References
Brown, L.G., Douglas, R., Fillmore, P.: Unitary equivalence modulo the compact operators and extensions of $C^*$-algebras. In: Proceedings of a Conference on Operator Theory, pp. 58-128. Dalhousie University, Halifax (1973)
Brown, L.G., Douglas, R., Fillmore, P.: Extensions of $C^*$-algebras and $K$-homology. Ann. Math. 105, 265-324 (1977)
Fang, X.: The classification of certain non-simple $C^*$-algebras of tracial rank zero. J. Funct. Anal. 256, 3861-3891 (2009)
Fang, X., Zhao, Y.: Approximately isometric lifting in quasidiagonal extensions. Sci. China 52(3), 457-467 (2009)
Fang, X., Zhao, Y.: The extensions of $C^*$-algebras with tracial topological rank no more than one. Ill. J. Math. 53(2), 441-462 (2009)
Fang, X., Zhao, Y.: The tracial topological rank of certain $C*$-algebras. Sci. China 54, 2295-2307 (2011)
Hu, S., Lin, H., Xue, Y.: The tracial topological rank of extensions of $C^*$-algebras. Math. Scand. 94, 125-147 (2004)
Lin, H.: An Introduction to the Classification of Amenable $C^*$-Algebras. World Scientific, Singapore (2001)
Lin, H.: Classification of simple TAF $C^*$-algebras. Can. J. Math. 53, 161-194 (2001)
Lin, H.: The tracial topological rank of $C^*$-algebras. Proc. Lond. Math. Soc. 83, 199-234 (2001)
Lin, H.: Tracially quasidiagonal extensions. Can. Math. Bull. 46, 388-399 (2003)
Lin, H.: Simple nuclear $C^*$-algebras of tracial topological rank one. J. Funct. Anal. 251, 601-679 (2007)
Lin, H.: Asymptotic unitary equivalence and classification of simple amenable $C^*$-algebras. Invent. Math. 364, 183-450 (2011)
Lin, H.: Approximate unitary equivalence in simple $C^*$-algebras of tracial rank one. Trans. Am. Soc. 364, 2021-2086 (2012)
Lin, H.: Approximately diagonalizing matrices over $C(Y)$. Proc. Natl. Acad. Sci. USA 109, 2842-2847 (2012)
Lin, H., Rørdam, M.: Extensions of inductive limits of circle $C^*$-algebras. J. Lond. Math. Soc. 51, 603-613 (1995)
Loring, T.A.: The noncommutative topology of one-dimensional spaces. Pac. J. Math. 136(1), 145-158 (1989)
Loring, T.A.: Lifting Solutions to Perturbing Problems in $C^*$-Algebras, vol. 8. Fields Institute Monographs, AMS, Providence (1997)
Acknowledgments
The authors are eternally grateful to Professor Huaxin Lin for his helpful lectures and very inspiring conversation during his staying at Shanghai.
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Communicated by Scott McCullough.
The first author was supported by ZJNSFC (No. LY13A010021) and the second author was supported by NNSFC (Nos. 11071188, 11371279).
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Zhao, Y., Fang, X. The Tracial Topological Rank of Extensions of \(C^*\)-Algebras. Complex Anal. Oper. Theory 10, 1181–1201 (2016). https://doi.org/10.1007/s11785-015-0488-1
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DOI: https://doi.org/10.1007/s11785-015-0488-1