1o Let N be a type II factor with the canonical trace . We call it a factor of the Haagerup type if there exists a net (P.). of normal linear maps on N which satisfy the following conditions(1) each P. is completely positive on N, (2) each P. is compact (i.e. for any 0, there exists a finite dimensional linear map Q on N such that P(x)-Q(x)[1x I[ for all xeN), and(3) P.(x)-x I]-.0, for all x e N.Here, we put I I x II= r(x*x) / for x e N.

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Abstract.Let G be a countable group which is not inner amenable. Then the II ,-factor M is full in the following cases:(1) M is given by the group measure space construction from a triple ( X, ¡i, G) with respect to a strongly ergodic measure preserving action of G on a probability space (A", ß).(2) M is the crossed product of a full 11,-factor by G with respect to an action.

Abstract.Fora *-endomorphism rj of an injective finite von Neumann algebra A , we investigate the relations among the entropy H(o) for a , the relative entropy H(A\a(A)) of a(A) for A , the generalized index k(A, cr(A)), and the index for subfactors. As an application, we have the following relations for the canonical shift T for the inclusion N c M of type II ( factors with the finite index [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 C M has finite depth, then all of them coincide.

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