High-dimensional Z-stable AH algebras

TL;DR

This paper investigates the structure of certain high-dimensional C*-algebras, showing they cannot be approximated by lower-dimensional subalgebras, contrasting with known dimension-reduction phenomena in simpler cases.

Contribution

It demonstrates that C(X,U) algebras with U UHF cannot be expressed as inductive limits of lower-dimensional subhomogeneous algebras, highlighting limitations in dimension reduction.

Findings

C(X,U) algebras are not inductive limits of lower-dimensional subhomogeneous algebras.

Contrasts with simple inductive limits where low-dimensional approximation is possible.

Dimension measured by decomposition rank can be low, but topological dimension remains high.

Abstract

It is shown that a C*-algebra of the form C(X,U), where U is a UHF algebra, is not an inductive limit of subhomogeneous C*-algebras of topological dimension less than that of X. This is in sharp contrast to dimension-reduction phenomenon in (i) simple inductive limits of such algebras, where classification implies low-dimensional approximations, and (ii) when dimension is measured using decomposition rank, as the author and Winter proved that dr(C(X,U)) $\leq$ 2.