# Journal of Operator Theory

Volume 71, Issue 1, Winter 2014 pp. 15-44.

MF-traces and a lower bound for the topological free entropy dimension in unital $C^*$-algebras**Authors**: Qihui Li (1), Don Hadwin (2), Weihua Li (3), and Junhao Shen (4)

**Author institution:**(1) Department of Mathematics, East China University of Science and Technology, Shanghai, China

(2) Department of Mathematics, University of New Hampshire, Durham, New Hampshire, U.S.A.

(3) Department of Science and Mathematics, Columbia College of Chicago, Chicago, IL, U.S.A.

(4) Department of Mathematics, University of New Hampshire, Durham, New Hampshire, U.S.A.

**Summary:**We continue the work on topological free entropy dimension $\delta_{\mathrm{top}}$ from D. Hadwin, Q. Li, J. Shen, Topological free entropy dimensions in nuclear $C^*$-algebras and in full free products of $C^*$-algebras, \textit{Canad. J. Math.} \textbf{63}(2011), 551--590, D. Hadwin, J. Shen, Topological free entropy dimension in unital $C^*$-algebras, \textit{J. Funct. Anal.} \textbf{256}(2009), 2027--2068, and D. Hadwin, J. Shen, Topological free entropy dimension. II, revised version. We introduce the notions of MF-trace, MF-ideal, and MF-nuclearity and use these concepts to obtain upper and lower bounds for $\delta_{\mathrm{top}}$, and in many cases we obtain an exact formula for $\delta_{\mathrm{top}}$. We also discuss semicontinuity properties of $\delta_{\mathrm{top}}$.

**DOI:**http://dx.doi.org/10.7900/jot.2011oct16.1951

**Keywords:**topological free entropy dimension, $C^*$-algebra, noncommutative continuous function, free product, MF-trace

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