On the homotopy equivalence of simple AI-algebras

Let $A$ and $B$ be simple unital AI-algebras (an AI-algebra is an inductive limit of $C^*$-algebras of the form $\bigoplus_i^kC([0,1],M_{N_i})$. It is proved that two arbitrary unital homomorphisms from $A$ into $B$ such that the corresponding maps $\mathrm K_0A\to\mathrm K_0B$ coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for $A$ and $B$ to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same $\mathrm K$-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, which is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix $h$ can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with $h$.