Geodesic
An Index Theory for Semigroups of *-Endomorphisms of and Type II1 Factors.
Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 86-114

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In this paper we study unit preserving *-endomorphisms of and type II1 factors. A *-endomorphism α which has the property that the intersection of the ranges of αn for n = 1 , 2 , ... consists solely of multiples of the unit are called shifts. In Section 2 it is shown that shifts of can be characterized up to outer conjugacy by an index n = ∞ 1, 2 , .... In Section 3 shifts of R the hyperfinite II1 factor are studied. An outer conjugacy invariant of a shift of R is the Jones index [R: α(R)]. In Section 3 a class of shifts of index 2 are studied. These are called binary shifts. It is shown that there are uncountably many binary shifts which are pairwise non conjugate and among the binary shifts there are at least a countable infinity of shifts which are pairwise not outer conjugate. 
Powers, Robert T. An Index Theory for Semigroups of *-Endomorphisms of and Type II1 Factors.. Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 86-114. doi: 10.4153/CJM-1988-004-3
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