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Thus n ~ A by Theorem 11. Now let (X, X, //, 5) be a dynamical system. , (Usf)(x) = f(Sx) for x £ X and / G L 2 ( M ). Definition 16. Let (XJ,XJ,(J,J,SJ) (j = 1,2) be a pair of dynamical systems with measure algebras 33i and «B2, respectively. Then, 5 i and 5 2 are said to be conjugate, denoted S\ ~ 52, if there exists a measure preserving onto isomorphism T : 93! ->

This means that U : r ( / n ) -> r ( / i 2 ) . The properties (1) - (5) are easily verified. 4. Algebraic models 33 The properties (1) and (5) in Proposition 4 are particularly important and will be used to characterize the conjugacy of measures in terms of their algebraic models. For this goal we need several technical results. Here, a pair of probability measure spaces (Xj,Xj,fij) {j = 1,2) are fixed. Lemma 5. Let V : L2(m) -> L2(fi2) be a linear isometry. If f E L°°{(i{) is such that {Vf)n = Vfn for n > 1, then Vf € L°°(/i 2 ) and ||V/||oo = l l / I U Proof.

For a general £ £ M s r (X) write f = f + - f" with £ + , £ - e M + ( X ) . Then we see that < H(iei,2i,5) < J|l||ifin = Mjien. \H(Z,%S)\ as desired. Lemma 5. if(-,2l, 5) is well-defined and is a bounded linear functional on such that \H(t,%s)\<\a\U\\, eeMs(x). 6) Proof. 1) that H(iZ,%S) = iH(Z,2L,S), H(i£,%S) iH(Z,%S), Hence it is easily seen that H(-,QL,S) is well-defined and linear by Lemmas 1, 2, 3 and 4. To see the boundedness of H(•, 2t, 5) note that for £, n e M J ( X ) max (||e||, 117,11) < U + iv\\ < VUW2 + IMP < 2||£ + : Then, we obtain \H(Z + ir,,%,S)\ = V\m,%,S)\2