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

http://dx.doi.org/10.4153/CJM-2011-014-8
Canad. J. Math. 63(2011), 551-590

  Published:2011-03-04
 Printed: Jun 2011

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.