By I. E. Segal
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Extra info for Decompositions of Operator Algebras I and II (Memoirs of the American Mathematical Society)
It results that WJ"T = J"TJ"WJ", and multiplying on the left by J", it follows that J"WJ" commutes with T. d = I. As The equation J'2 = I implies that J12(x4y) = x#y for all x and y in k, and DECOMPOSITIONS OF OPERATOR ALGEHRAS. 57 I substituting J' = (V#I)C' there results the equation (Vx)#J"VJ"y) = x#y. Hence V = I and J' = C'. The proof of the lemma is concluded by the observ- ation that as JL(a)J = h(a), a e G, we have h"(A) = W(L"(a)) so that R"(a) = J"L"(a)J". 10. Deflation of decompositions.
2,,(V), `9,1(U*) 1>(W)) dit( 7l) _ (Ty) ,(V), (UaW) I) (TVz, U*Wz) = (UTVz, Wz). V As and W range over Q , Vz range over dense subsets of AP, and from this it follows that _ (UTx, y) for all x and y in W', so that (i) and Wz (TUx, y) TU = UT. T E Q'^ C; but by assumption, Q'n C - C. Now let T( -61) be the continuous function on f corresponding to T and let S be arbitrary in C . 1) - fS(2()(Tax( I), y( a()) dj ( I). On the Thus 32 I. E. &( j). e. e. 14/(U1) are dense in are both states, PhOOF O6 THEOhEtd.
In view of the known correspondence between positive definite functions on groups and ccntinuous unitary representations of groups , our result generalizes the representation theorem for positive definite functions on locally compact abelian groups by showing that on a separable locally compact group, every measurable positive definite function can be represented as an integral of "elementary" positive definite function, where an "ele- mentary" function is defined as one which is not a nontrivial convex linear combination of two other such functions (or alternatively, as one for which the associated group representation is Irreducible).