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

数学

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

Let \epsilon>0. Is there \delta>0 satisfying the following? Given any pair of unitaries u and v in a unital separable simple C*-algebra A for which  \|uv-vu\|<\delta, there is a continuous path of unitaries \{v(t): t\in [0,1]\}\subset A such that  v(0)=v, v(1)=1 and \|uv(t)-v(t)u\|<\epsilon for all t\in [0,1]. We give an answer to this question when A is assumed to be a unital simple C*-algebra with tracial rank no more than one.
Let C be a unital separable simple C*-algebra, with tracial rank no more than one. Suppose that \phi: C\to A is a unital monomorphism and u\in A is a unitary such that \phi almost commutes with u. We give an answer to the question when there is a continuous path of unitaries \{u(t): t\in [0,1]\} with u(0)=u and u(1)=1 such that entire path u(t) almost commute with \phi. Other versions of so-called basic homotopy lemma are also presented.