Deformation Retraction of Groups and Toeplitz Algebras

Abstract:

      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.