## Abstract

Suppose that a simple *C**-algebra A can be written as an inductive limit of , where *X _{n,i}* are finite CW complexes with uniformly bounded dimensions. Such inductive limit

*C**-algebras are called AH algebras.

In this paper, we will prove that for any *n* and *ε* > 0, there exists a homomorphism *ψ _{n,m}*:

*A*→

_{n}*A*, which factors through an interval algebra , such that the spectra of

_{m}*ψ*and the spectra of

_{n,m}*φ*can be paired within

_{n,m}*ε*provided that

*m*is large enough. This result is essential for the classification of such simple

*C**-algebras.

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