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arXiv 1301.0356math.OAmath.FA
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Z-稳定单C* 代数-代数的指数秩和指数长度

Exponential rank and exponential length for Z-stable simple C*-algebras

Huaxin Lin

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摘要：

设$A$是有理迹秩至多为1的酉可分单${\cal Z}$ -稳定C*-代数，$u\in U_0(A),$是$A.$的酉群的连通分支，证明了对任意$ε>0,$，存在自伴元$h\in A$，使得$$ |u-\exp(ih)|<ε. $$ $|h|$的下界可以任意大.若$u\in CU(A),$是酉群的换位子群的闭包，则证明了存在自伴元$h\in A$，使得$$ |u-\exp(ih)| <εand |h|\le 2π. $$.给出了$|h|$的界$2π$一般是最优的例子.对于江苏代数${\cal Z},$，我们证明了：如果$u\in U_0({\cal Z})$和$ε>0,$存在实数$-π<t\le π$和具有$|h|\le 2π$的自伴元$h\in {\cal Z}$，使得$$ |e^{it}u-\exp(ih)|<ε. $$

英文摘要：

Let $A$ be a unital separable simple ${\cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $u\in U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $ε>0,$ there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)|<ε. $$ The lower bound of $|h|$ could be as large as one wants. If $u\in CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)| <εand |h|\le 2π. $$ Examples are given that the bound $2π$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${\cal Z},$ we show that, if $u\in U_0({\cal Z})$ and $ε>0,$ there exists a real number $-π<t\le π$ and a self-adjoint element $h\in {\cal Z}$ with $|h|\le 2π$ such that $$ |e^{it}u-\exp(ih)|<ε. $$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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