# Huaxin Lin: Homotopy of unitaries in simple C*-algebras with tarcial rank one. (arXiv:0805.0583v2 [math.OA] UPDATED)

[0805.0583] Homotopy of unitaries in simple C*-algebras with tracial rank one

Let . Is there satisfying the following? Given any pair of unitaries and in a unital separable simple C*-algebra for which , there is a continuous path of unitaries such that , and for all . We give an answer to this question when is assumed to be a unital simple C*-algebra with tracial rank no more than one.
Let be a unital separable simple C*-algebra, with tracial rank no more than one. Suppose that is a unital monomorphism and is a unitary such that almost commutes with . We give an answer to the question when there is a continuous path of unitaries with and such that entire path almost commute with . Other versions of so-called basic homotopy lemma are also presented.