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Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras

Published:2011-03-04
Printed: Jun 2011
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A.
• Qihui Li,
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A.
• Junhao Shen,
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A.
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Abstract

In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C$^{\ast}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that $\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group.
 Keywords: topological free entropy dimension, unital C$^{*}$-algebra
 MSC Classifications: 46L10 - General theory of von Neumann algebras 46L54 - Free probability and free operator algebras

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