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arXiv · 1010.0623

Mean dimension and AH-algebras with diagonal maps

Zhuang Niu
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Abstract

Mean dimension for AH-algebras is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the K0-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero).

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Zhuang Niu. 2010-10-04. Mean dimension and AH-algebras with diagonal maps. https://arxiv.org/abs/1010.0623

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