By Dr. Şerban-Valentin Strâtilâ, Dr. Dan-Virgil Voiculescu (auth.)

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17. In this Section a general method of constructing - representations of A(i~, P ) outlined . Unfortunately and hence also of A , will be , it is of limited use for our purposes and 53 since it is difficult to decide when such a representation factor representation For is a . f ~ C(I-L) and ~ P let us denote by f~ C(~) the function defined by f~(t) Then , if ~ = f(r is a representation of space H , we have that ~ ~ of and is a r Moreover , f~ ~(Tf) t ~ ~ A(~,~) >~(V~) . on some Hilbert is a unitary representation * - representation of C(A~-) .

5. ~ is now an easy consequence applied to the restriction 45 of z to Z~(c)" = Conversely L~(fl,~) , let tive measure on ~-L ~ . be a sigma-finite , equivalent to ~ [~ - invariant posi- . For x e ~(A)" , x >i 0, we define ~(x) = (JfLP~(x)(t) dg(t) which is correctly defined since with respect to That Z ~ W is absolutely continuous . , (9) and ([2)) , ~ continuous with respect to 9 a direct consequence To prove that Consider ej t g . Also , that of the fact that ~ ~ -g being absolutely is semifinite is sigma-finite is .

UC) = A . D. 3. We shall prove that the representation standard , more precisely we have PROPOSITION measure on ~ Proof . Le__~t ~ . Then ~ . B y the Suppose x ~ UTp(A)" density theorem ~uasi-invariant probability construction ~ ~ is separating is such that xk g A is cyclic . For A SI~(A)" we have Kaplansky there is a norm-boun- such that ~IV~(xk) converges x . (UC) . , so all x g A x ~ p = 0 . By the and the separability of ded sequence of elements strongly to P- beg ~ is : is c2clic and s0parating for GNS we have to prove is that 71~ H~.