Homotopy of unitaries in simple C*-algebras with tracial rank one
| dc.creator | Lin, Huaxin | |
| dc.date | 2008-05-05 | |
| dc.date | 2009-05-21 | |
| dc.date.accessioned | 2026-07-07T13:16:37Z | |
| dc.date.available | 2026-07-07T13:16:37Z | |
| dc.description | 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 $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<ε\forall t\in [0,1]. $$ 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 $ϕ: C\to A$ is a unital monomorphism and suppose that $v\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $ϕ.$ It is shown that there is a continuous path of unitaries $\{v(t): t\in [0,1]\}$ 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. | |
| dc.description | 50 pages | |
| dc.identifier | https://arxiv.org/abs/0805.0583 | |
| dc.identifier | http://arxiv.org/abs/0805.0583 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230850 | |
| dc.subject | Operator Algebras | |
| dc.subject | Functional Analysis | |
| dc.subject | 46L35, 46L80 | |
| dc.title | Homotopy of unitaries in simple C*-algebras with tracial rank one | |
| dc.type | text |