Powers’s binary shifts on the hyperfinite factor of type $\textrm {II}_ 1$

Abstract:

A unit preserving $*$-endomorphism $\sigma$ on the hyperfinite ${\text {I}}{{\text {I}}_1}$ factor $R$ is called a shift if $\bigcap \nolimits _{n = 0}^\infty {{\sigma ^n}(R) = \{ \lambda 1;\lambda \in \mathbb {C}} \}$. A shift $\sigma$ is called Powers’ binary shift if there is a self-adjoint unitary $u$ such that $R = \{ {\sigma ^n}(u);n \in \mathbb {N} \cup \{ 0\} \} ''$ and ${\sigma ^k}(u)u = \pm u{\sigma ^k}(u)$ for $k \in \mathbb {N} \cup \{ 0\}$. Let $q(\sigma )$ be the number $\min \{ k \in \mathbb {N};{\sigma ^k}(R)’ \cap R \ne \mathbb {C}1\}$. It is shown that the number $q(\sigma )$ is not the complete outer conjugacy invariant for Powers’ binary shifts.