When do triple operator integrals take value in the trace class?

Consider three normal operators A, B, C on a separable Hilbert space H as well as scalar-valued spectral measures lambda(A) on sigma(A), lambda(B) on sigma(B) and lambda(C) on sigma(C). For any phi is an element of L-infinity (lambda(A) X lambda(B) X lambda(C)) and any X, Y is an element of S-2 (H), the space of Hilbert-Schmidt operators on H, we provide a general definition of a triple operator integral Gamma (A,B,C) (phi)(X, Y) belonging to S-2(H) in such a way that Gamma(A,B,C) (phi) belongs to the space B-2(S-2 (H) x S-2 (H) , S-2 (H)) of bounded bilinear operators on S-2 (H), and the resulting mapping Gamma(A,B,C): L-infinity (lambda(A) X lambda(B) X lambda(C)) -> B-2 (S-2(7H) x S-2 (H), S-2(H)) is a omega*-continuous isometry. Then we show that a function phi is an element of L-infinity (lambda(A) x lambda(B) x lambda(C)) has the property that Gamma(A,B,C) (phi) maps S-2 (H) x S-2 (H) into S-1 (H), the space of trace class operators on H, if and only if it has the following factorization property: there exist a Hilbert space H and two functions a is an element of (lambda(A) x lambda(B); H) and b is an element of L-infinity (lambda(B) x lambda(C); H) such that phi(t(1), t(2), t(3)) = < a(t(1) , t(2)), b(t(2), t(3))> for a.e. (t(1), t(2), t(3)) is an element of sigma (A)x sigma - (B) x sigma (C). This is a bilinear version of Peller's Theorem characterizing double operator integral mappings S-1(H) -> S-1 (H). In passing we show that for any separable Banach spaces E, F, any omega-*measurable esssentially bounded function valued in the Banach space Gamma(2) (E, F*) of operators from E into F* factoring through Hilbert space admits a w*-measurable Hilbert space factorization.