Abstract
In the paper, we consider the question as to whether a unital full amalgamated free product of quasidiagonal C*-algebras is itself quasidiagonal. We give a sufficient condition for a unital full amalgamated free product of quasidiagonal C*-algebras with amalgamation over a finite-dimensional C*-algebra to be quasidiagonal. By applying this result, we conclude that the unital full free product of two AF algebras with amalgamation over a finite-dimensional C*-algebra is AF if there exists a faithful tracial state on each of the two AF algebras such that the restrictions of these states to the common subalgebra coincide.
Similar content being viewed by others
References
S. Armstrong, K. Dykema, R. Exel, and H. Li, “On embeddings of full amalgamated free product C*-algebras”, Proc. Amer. Math. Soc., 132:7 (2004), 2019–2030.
F. Boca, “A note on full free product C*-algebras, lifting and quasidiagonality,” in: Operator Algebras and Related Topics (Proceedings of the 16th Operator Theory Conference, Timi¸soara, 1996), Theta Found., Bucharest, 1997, 51–63.
N. P. Brown and K. J. Dykema, “Popa algebras in free group factors”, J. Reine Angew. Math., 573 (2004), 157–180.
F. Boca, “Completely positive maps on amalgamated product C*-algebras”, Math. Scand., 72:2 (1993), 212–222.
B. Blackadar and E. Kirchberg, “Generalized inductive limits of finite-dimensional C*-algebras”, Math. Ann., 307:3 (1997), 343–380.
N. P. Brown, “On quasidiagonal C*-algebras”, in: Operator Algebras and Applications, Advanced Studies in Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, 19–64.
N. P. Brown and N. Ozawa, C*-Algebras and Finite-Dimensional Approximations, Amer. Math. Soc., Providence, RI, 2008.
K. Davidson, C*-Algebras by Example, Amer. Math. Soc., Providence, RI, 1996.
R. Exel and T. Loring, “Finite-dimensional representations of free product C-algebras”, Internat. J. Math., 3:4 (1992), 469–476.
D. Hadwin, “Nonseparable approximate equivalence”, Trans. Amer. Math. Soc., 266:1 (1981), 203–231.
D. Hadwin, “A lifting characterization of RFD C*-algebras”, Math. Scand., 115:1 (2014), 85–95.
D. Hadwin, Q. Li, and J. Shen, “Topological free entropy dimensions in nuclear C*-algebras and in full free products of unital C*-algebras”, Canad. J. Math., 63 (2011), 551–590.
U. Haagerup and S. Thorbjørnsen, “A new application of random matrices: Ext(C*red(F 2)) is not a group,” Ann. of Math. (2), 162:2 (2005), 711–775.
R. Kadison and J. Ringrose, Fundamentals of the Operator Algebras, vol. 1, 2, Academic Press, Orlando, FL, 1983, 1986.
Q. Li and J. Shen, “A note on unital full amalgamated free products of RFD C*-algebras”, Illinois J. Math, 56:2 (2012), 647–659.
Q. Li and J. Shen, “Unital full amalgamated free products of MF C*-algebras”, Oper. Matrices, 7:2 (2013), 333–356.
T. Loring, Lifting Solutions to Perturbing Problems in C-algebras, Amer. Math. Soc., Providence, RI, 1997.
G. K. Pedersen, “Pullback and pushout constructions in C*-algebra theory”, J. Funct. Anal., 167:2 (1999), 243–344.
D. Voiculescu, “A note on quasi-diagonal C*-algebras and homotopy”, Duke Math. J., 62:2 (1991), 267–271.
D. Voiculescu, “Around quasidiagonal operators”, Integral Equations Operator Theory, 17:1 (1993), 137–149.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author is partially supported by National Natural Science Foundation of China (Grant No. 11201146) and the Fundamental Research Funds for the Central Universities as well as SRF for ROCS, SEM.
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 50, No. 1, pp. 38–46, 2016
Rights and permissions
About this article
Cite this article
Li, Q., Hadwin, D., Li, J. et al. On Unital Full Amalgamated Free Products of Quasidiagonal C*-Algebras. Funct Anal Its Appl 50, 39–47 (2016). https://doi.org/10.1007/s10688-016-0126-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-016-0126-3