Author’s abstract: Suppose that a simple \(C^*\)-algebra \(A\) can be written as an inductive limit of \((A_n= \bigoplus^{k_n}_{i= 1} M_{[n,i]}(C(X_{n,i})), \varphi_{n,m})\), 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 \(\varepsilon>0\), there exists a homomorphism \(\Psi_{n,m}: A_n\to A_m\), which factors through an interval algebra \(\bigoplus^{k_n}_{i= 1}M_{[n,i]} (C[0,1])\), such that the spectra of \(\Psi_{n,m}\) and the spectra of \(\varphi_{n,m}\) can be paired within \(\varepsilon\) provided that \(m\) is large enough. This result is essential for the classification of such simple \(C^*\)-algebras.