Bundle convergence of sequences of pairwise uncorrelated operators in von Neumann algebras and vectors in their L2-spaces ⋆

Let H be a separable complex Hilbert space, A a von Neumann algebra in ℒ(H),a faithful, normal state on A. We prove that if a sequence (Xn: n ≥ 1) of uncorrelated operators in A is bundle convergent to some operator X in A and Σ∞n=1n−2 Var(Xn) log2(n + 1) < ∞, then X is proportional to the identity operator on H. We also prove an analogous theorem for certain uncorrelated vectors in the completion L2=L2(A,φ) of A given by the Gelfand-Naimark-Segal representation theorem. Both theorems were motivated by a recent one proved by Etemadi and Lenzhen in the classical commutative setting.