群的形变收缩及Toeplitz代数

设G为一个离散群,(G,G_ )为一个拟偏序群使得G_ ~0=G_ ∩G_ ~(-1)为G的非平凡子群。令G]为G关于G_ ~0的左倍集全体,|G_ |为G]的正部。记T~(G_ )和T~(G_ ])为相应的Toeplitz代数。当存在一个从G到G_ ~0上的形变收缩映照时,我们证明了T~(G_ )酉同构于T~(G_ ])×C_r~*(G_ ~0)的一个C_-~*c子代数。若进一步,G_ ~0还为G的一个正规子群,则T~(G_ )与T~(G_ ])×C_r~*(G_ ~0)酉同构。

Deformation Retraction of Groups and Toeplitz Algebras

Let $(G,G_+)$ be a quasi-partial ordered group such that $G_+^0=G_+\cap G_+^{-1}$ is a non-trivial subgroup of $G$. Let $G]$ be the collection of left cosets and $G_+]$ be its positive. Denote by ${\cal T}^{G_+}$ and ${\cal T}^{G_+]}$ the associated Toeplitz algebras. We prove that ${\cal T}^{G_+}$ is unitarily isomorphic to a $C^*$-subalgebra of ${\cal T}^{G_+]}\otimes C_r^*(G_+^0)$ if there exists a deformation retraction from $G$ onto $G_+^0$. Suppose further that $G_+^0$ is normal, then ${\cal T}^{G_+}$ and ${\cal T}^{G_+]}\otimes C_r^*(G_+^0)$ are unitarily equivalent.