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1. CJM 2014 (vol 67 pp. 369)
A Free Product Formula for the Sofic Dimension It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$-regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.
Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy Category:20E06 |
2. CJM 2011 (vol 63 pp. 551)
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras |
Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras 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 Categories:46L10, 46L54 |