ENTROPY FOR CANONICAL SHIFTS .2.

Let N subset-of M be a pair of factors. Associated with a conditional expectation E from M onto N with finite index, we introduce the canonical shift-GAMMA on the von Neumann algebra A, with the canonical state-phi, generated by the tower of relative commutants for the basic constructions iterated from E. Related with the minimum index [M:N]0, we investigate the entropy h-phi(GAMMA) of GAMMA and the entropy H-phi(A\GAMMA(A)) of A relative to the subalgebra-GAMMA(A). The inequalities h-phi(GAMMA) less-than-or-equal-to log[M:N]0 and 1/2 H-phi(A\GAMMA(A)) less-than-or-equal-to log[M:N]0 hold in general. Furthermore when E has the minimum index and N subset-of M has finite depth, we establish h-phi(GAMMA) = 1/2 H-phi(A\GAMMA(A)) = log [M:N]0.