## Powersโs binary shifts on the hyperfinite factor of type $\textrm {II}_ 1$

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- by Masatoshi Enomoto and Yasuo Watatani
- Proc. Amer. Math. Soc.
**105**(1989), 371-374 - DOI: https://doi.org/10.1090/S0002-9939-1989-0938911-6
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## 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 Powersโ 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 $\min \{ k \in \mathbb {N};{\sigma ^k}(R)โ \cap R \ne \mathbb {C}1\}$. It is shown that the number $q(\sigma )$ is not the complete outer conjugacy invariant for Powersโ binary shifts.## References

- D. Bures and H. S. Yin,
- Marie Choda,
*Shifts on the hyperfinite $\textrm {II}_1$-factor*, J. Operator Theory**17**(1987), no.ย 2, 223โ235. MR**887220**
M. Enomoto, M. Choda and Y. Watatani, - V. F. R. Jones,
*Index for subfactors*, Invent. Math.**72**(1983), no.ย 1, 1โ25. MR**696688**, DOI 10.1007/BF01389127 - Robert T. Powers,
*An index theory for semigroups of $^*$-endomorphisms of ${\scr B}({\scr H})$ and type $\textrm {II}_1$ factors*, Canad. J. Math.**40**(1988), no.ย 1, 86โ114. MR**928215**, DOI 10.4153/CJM-1988-004-3 - Geoffrey L. Price,
*Shifts on type $\textrm {II}_1$ factors*, Canad. J. Math.**39**(1987), no.ย 2, 492โ511. MR**899846**, DOI 10.4153/CJM-1987-021-2 - Geoffrey L. Price,
*Shifts of integer index on the hyperfinite $\textrm {II}_1$ factor*, Pacific J. Math.**132**(1988), no.ย 2, 379โ390. MR**934178**, DOI 10.2140/pjm.1988.132.379

*Shifts on the hyperfinite factor of type*${\text {I}}{{\text {I}}_1}$ (preprint, 1987).

*Uncountably many non-binary shifts on the hyperfinite*${\text {I}}{{\text {I}}_1}$

*-factor*(preprint, 1987).

## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**105**(1989), 371-374 - MSC: Primary 46L10; Secondary 46L35, 46L55
- DOI: https://doi.org/10.1090/S0002-9939-1989-0938911-6
- MathSciNet review: 938911