Non-Z-Stable Simple AH Algebras

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.