Unitaries in a Simple C*-algebra of Tracial Rank One

2009/01/30 by Huaxin Lin, Lin, Huaxin
Mathematics · #46L05 #46L35 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #math.OA #msc:46L05 #msc:46L35

paper · pdf · doi:10.48550/arxiv.0902.0024

arxiv created 2009/01/30 · openalex publication_date 2009/01/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be a unital separable simple infinite dimensional \CA with tracial rank no more than one and with the tracial state space T(A) and let U(A) be the unitary group of A. Suppose that u∈ U₀(A), the connected component of U(A) containing the identity. We show that, for any \ep>0, there exists a selfadjoint element h∈ A_s.a such that ‖u-exp(ih)‖<\ep. We also study the problem when u can be approximated by unitaries in A with finite spectrum. Denote by CU(A) the closure of the subgroup of unitary group of U(A) generated by its commutators. It is known that CU(A)⊂ U₀(A). Denote by \widehata the affine function on T(A) defined by \widehata(τ)=τ(a). We show that u can be approximated by unitaries in A with finite spectrum if and only if u∈ CU(A) and \widehatuⁿ+(uⁿ)^*,i(\widehatuⁿ-(uⁿ)^*)∈ ρ_A(K₀(A) for all n≥ 1. Examples are given that there are unitaries in CU(A) which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.

Citations

• Homotopy of unitaries in simple C*-algebras with tracial rank one

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