This paper is a natural continuation of the previous paper [9]. In [9], we proved a structure theorem for a von Neumann algebra with a fixed periodic and homogeneous state. In this paper, we will show that the structure theorem in [9] determines intrinsically the algebraic type of a factor with a periodic and inner homogeneous state (see Definition 1). We keep the terminologies and the notations in [9]. DEFINITION 1. A normal state cp on a von Neumann algebra^is said to be inner homogeneous if

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... {cp) n Int (M) acts ergodically on M, that is, if the group of all inner automorphisms of M leaving cp invariant has no fixed points other than the scalar multiples of the identity. For each a e M, we write Ad(a)x = axa*, xeM. Since Ad(u)eG((p) for a unitary ueM if and only if u falls in M^ the centralizer of cp, the inner homogeneity of cp is equivalent to the fact that M'y n M = {Al}. Hence M^ is a II r factor and M itself is also a factor. We consider two periodic and inner homogeneous faithful normal states cp and \j/ on M. We denote by {M* : n = 0, ± 1,...} and {Mf : n = 0, ± 1,...} the decompositions of M in [9, Theorem 11] corresponding to cp and xjj respectively. By [9, Theorem 13], cp and \j/ have the same period, say T > 0. Let K = e~2 n/T , 0 < K < 1. Following Connes' idea, we consider the tensor product & = M (x) i?( § 2 ) of M and the 2 x 2-matrix algebra JS?($ 2 )-Let {e itj : ij = 1, 2} be a system of matrix units in J£?(ô 2 )-Every x e 3P is of the form x = x n (g) e ix + x 12 e 12 + x 21 e 21 + x 22 ® e 22 , where x y e ^. We define a faithful state % on SP by ZW = i(<P(Xii) + <M*22))-Connes showed in [3] that there exists a strongly continuous one-AMS (MOS) subject classifications (1970). Primary 46L10.