2008/05/05 by Huaxin Lin, Lin, Huaxin · 2 citations
Mathematics · #46L35 #46L80 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L35 #msc:46L80
paper · pdf · doi:10.48550/arxiv.0805.0583
50 pages
openalex publication_date 2008/05/05 · arxiv created 2009/05/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ε>0 be a positive number. Is there a number δ>0 satisfying the following? Given any pair of unitaries u and v in a unital simple C^*-algebra A with [v]=0 in K1(A) for which ‖uv-vu‖<\dt, there is a continuous path of unitaries \v(t): t∈ [0,1]\⊂ A such that v(0)=v, v(1)=1 \and ‖uv(t)-v(t)u‖<ε∀ t∈ [0,1]. An answer is given 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 amenable simple C^*-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that ϕ: C→ A is a unital monomorphism and suppose that v∈ A is a unitary with [v]=0 in K1(A) such that v almost commutes with ϕ. It is shown that there is a continuous path of unitaries \v(t): t∈ [0,1]\ in A with v(0)=v and v(1)=1 such that the entire path v(t) almost commutes with ϕ, provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented.