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 ⊗^k_i=1 C([0, 1], M_Ni)). It is proved that two arbitrary unital homomorphisms from A into B such that the corresponding maps K₀A → K₀B 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 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 Manuǐlov 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.