Non-Z-Stable Simple AH Algebras
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Date
2024-01-09
Authors
Hendrickson, Allan
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Publisher
University of Oregon
Abstract
We consider the problem of dimension growth in AH algebras $A$ defined as inductive limits $A = \lim_{n \to \infty} (M_{R_n}(C(X_n)),\phi_{n})$ over finite connected CW-complexes $X_n$. We show that given any sequence $(X_n)$ of finite connected CW-complexes and matrix sizes $(R_n)$ with $R_n \rvert R_{n+1}$ satisfying the dimension growth condition $ \lim_{n \to \infty} \frac{\dim(X_n)}{R_n} = c$ with $c \in (0,\infty)$, there always exists an AH algebra with injective connecting homomorphisms over a subsequence which does not have Blackadar's strict comparison of positive elements, and therefore does not absorb tensorially the Jiang-Su algebra $Z$. This demonstrates that no regularity condition can be placed on the spaces $X_n$ in order to stabilize AH algebras over them - there always exists a pathological construction.
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Keywords
AH algebra, Elliott Invariant, Radius of comparison, Villadsen