曲面群的拟表示
Quasi-representations of surface groups
摘要：
所谓群$G$的拟表示，我们指的是$G$到单位$C^*$ -代数的酉群的近似乘法映射。一个拟表示在$K$ -理论的层次上导出一个部分定义的映射。在90年代早期，Exel和Loring将两个不变量关联到几乎可交换的酉矩阵对$u$和$v$：一个是$K$理论不变量，它可以被认为是$\mathbb{Z}^2$在U（n）中的拟表示所诱导的映射下$K_0(C(\mathbb{T}^2))$中的Bott元素的像;另一个是$\mathbb{C}\setminus \{0\}$中封闭路径$t\mapsto \det(tvu + (1-t)uv)$的缠绕数。所谓的Exel-Loring公式指出，如果$\|uv - vu\|$足够小，则这两个不变量重合。第二位作者给出了取值于U（n）的曲面群的拟表示的Exel-Loring公式的推广。本文将这个公式进一步推广到取值于迹单位$C^*$ -代数的酉群的曲面群的拟表示。
英文摘要：
By a quasi-representation of a group $G$ we mean an approximately multiplicative map of $G$ to the unitary group of a unital $C^*$-algebra. A quasi-representation induces a partially defined map at the level $K$-theory. In the early 90s Exel and Loring associated two invariants to almost-commuting pairs of unitary matrices $u$ and $v$: one a $K$-theoretic invariant, which may be regarded as the image of the Bott element in $K_0(C(\mathbb{T}^2))$ under a map induced by quasi-representation of $\mathbb{Z}^2$ in U(n); the other is the winding number in $\mathbb{C}\setminus \{0\}$ of the closed path $t\mapsto \det(tvu + (1-t)uv)$. The so-called Exel-Loring formula states that these two invariants coincide if $\|uv - vu\|$ is sufficiently small. A generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U(n) was given by the second-named author. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital $C^*$-algebra.
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