EXTENSION ALGEBRAS OF II1 FACTORS VIA *-ENDOMORPHISMS

Abstract

From the view of Jones index theory, we construct a III factor <A, γ>, for a pair {A, γ} of a II 1 factor A and a trace preserving *-endomorphism γ, through the method of the crossed product by a *-endomorphism. The AFD III λ factor M is isomorphic to <A, σ> for a basic *-endomorphism σ of the hyperfinite II 1 factor A if and only if λ−1 ∈ {4 cos 2(π/n : n ≥ 3} ⋃ [4, ∞). Ocneanu's canonical shift Γ for an inclusion N ⊂ M of II 1-factors with [M: N] < ∞ is extended to a *-endomorphism [Formula: see text] of <A, Γ> for A = M′ ∩ M∞. The extended *-endomorphism [Formula: see text] is Longo's canonical *-endomorphism.