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


Published:20110304
Printed: Jun 2011
Don Hadwin,
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.
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 selfadjoint 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.