ENTROPY FOR CANONICAL SHIFTS

For a *-endomorphism sigma of an injective finite von Neumann algebra A , we investigate the relations among the entropy H(sigma) for sigma , the relative entropy H(A\sigma(A)) of sigma(A) for A , the generalized index lambda(A, sigma(A)) , and the index for subfactors. As an application, we have the following relations for the canonical shift GAMMA for the inclusion N subset-of M of type II1 factors with the finite index [M: N], H(A\GAMMA(A)) less-than-or-equal-to 2H(GAMMA) less-than-or-equal-to log lambda(A, GAMMA(A))-1 = 2 log[M : N], where A is the von Neumann algebra generated by the two of the relative commutants of M. In the case of that N subset-of M has finite depth, then all of them coincide.