A Noncompletely Continuous Operator on L 1 (G) whose Random Fourier Transform is in c 0 (G)

Abstract

Let T be a bounded linear operator from Li(G, A) into Li(f!,7, P), where (G, A) is a compact abelian metric group with its Haar measure, and (f2, 7, P) is a probability space. Let (sw) be the random mea- sure on G associated to T; that is, T/(oj) = fG f(t) dpu(t) for each / in Li(G). We show that, unlike the ideals of representable and Kalton operators, there is no subideal B of M(G) such that T is completely continuous if and only if su> G B for almost u> in Q. We actually exhibit a noncompletely continuous operator T such that su G I2+e(G) for each e > 0. I. Random measures associated to operators on h\. Let K be a separable compact Hausdorff space, A a Radon probability on K and (U, f,P) a probability space. Denote by M(K) the space of all Radon measures on K. We will call random measure on K every measurable map w —+ p^ from (U, 7, P) into M(K) when M(K) is equipped with the cr-field generated by the o(M(K), C(K)) topology. The starting point of this paper is the following disintegration theorem estab- lished by Kalton (3) and Fakhoury (2). (a) If T is a bounded linear operator from Lx(K, X) into Li(fi, 7, P), then there exists an essentially unique random measure (pj) verifying the following properties: (i) Each / in Lx(K, X) belongs to Lx(K, \p^\) for P almost all uj. (ii) Tf(w) = iK f(t) dpu(t) for P almost all w. (iii) / |/ij dP(u) Li(f2)