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

Published online by Cambridge University Press:  20 November 2018

Don Hadwin ,
Qihui Li  and
Junhao Shen
Don Hadwin
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A. email: don@math.unh.eduqme2@cisunix.unh.edujog2@cisunix.unh.edu
Qihui Li
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A. email: don@math.unh.eduqme2@cisunix.unh.edujog2@cisunix.unh.edu
Junhao Shen
Affiliation:
Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, U.S.A. email: don@math.unh.eduqme2@cisunix.unh.edujog2@cisunix.unh.edu
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Abstract

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In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital ${{\text{C}}^{*}}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear ${{\text{C}}^{*}}$-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 ${{\text{C}}^{*}}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital $\text{MF}$ algebras is also an $\text{MF}$ algebra. As an application, we obtain that $\text{Ext(}C_{r}^{*}\text{(}{{F}_{2}}\text{)}{{\text{*}}_{\mathbb{C}}}C_{r}^{*}\text{(}{{F}_{2}}\text{))}$ is not a group.

Keywords

46L1046L54topological free entropy dimensionunital C*-algebra
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 3 , 01 June 2011 , pp. 551 - 590
Copyright
Copyright © Canadian Mathematical Society 2011

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