Shifts on Type <i>II</i>₁ Factors

Abstract

A shift on a unital C *-algebras is a *-endomorphism α of which fixes the identity and has the property that the intersection of the ranges of α n for n = 1, 2, 3, … consists only of multiples of the identity. In [ 4 ] R. T. Powers introduced the notion of a shift on a C *-algebra and considered both discrete and continuous one-parameter semi-groups of shifts. In this paper we focus on discrete shifts. We use a construction of Powers to obtain shifts on certain unital AF C *-algebras. These are defined by constructing a set { u i :i = 1, 2, …} of self-adjoint unitary operators which pairwise either commute or anticommute. Setting α(u i ) = u i + 1 , determines an endomorphism on the group algebra generated by the u i 's. This algebra is called a binary shift algebra. By passing to the (unique) C *-algebra completion we obtain an AF -algebra on which a defines a shift.