Unitaries in a Simple C*-algebra of Tracial Rank One
| dc.creator | Lin, Huaxin | |
| dc.date | 2009-01-30 | |
| dc.date.accessioned | 2026-07-07T12:36:48Z | |
| dc.date.available | 2026-07-07T12:36:48Z | |
| dc.description | 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 $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\ep>0,$ there exists a selfadjoint element $h\in A_{s.a}$ 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 $CU(A)\subset U_0(A).$ Denote by $\widehat{a}$ the affine function on $T(A)$ defined by $\widehat{a}(τ)=τ(a).$ We show that $u$ can be approximated by unitaries in $A$ with finite spectrum if and only if $u\in CU(A)$ and $\widehat{u^n+(u^n)^*},i(\widehat{u^n-(u^n)^*})\in \overline{ρ_A(K_0(A)}$ for all $n\ge 1.$ 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. | |
| dc.identifier | https://arxiv.org/abs/0902.0024 | |
| dc.identifier | http://arxiv.org/abs/0902.0024 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/218219 | |
| dc.subject | Operator Algebras | |
| dc.subject | 46L05, 46L35 | |
| dc.title | Unitaries in a Simple C*-algebra of Tracial Rank One | |
| dc.type | text |