Bundle convergence of weiteighted sums of operators in noncommutative [L_2]-spaces

The notion of bundle convergence for sequences of operators in a von Neumann algebra A equipped with a faithful and normal state phi as well as for sequences of vectors in their [L_2]-spaces were introduced by Hensz, Jajte and Paszkiewicz in 1996 as an appropriate substitute for almost everywhere convergence in the commutative setting. First, we prove that if (B_k : k = l, 2,...) is a sequence in A such that [sum{phi(B^*_k B_k)} < infinity] and if (a_k : k = l, 2,...) is a sequence of complex numbers such that [sum|a_k|^2 < infinity], then the *-homomorph image of the sequence [(sum_{k = 1}^n {a_k B_k : n = 1, 2,...})] in the [L_2] space given by the Gelfand-Naimark-Segal representation theorem is bundle convergent. Second, we prove a noncommutative version of the second Borel-Cantelli lemma in terms of bundle convergence. Such a version of the first Borel-Cantelli lemma cannot exist in this noncommutative setting. On closing, we raise two problems.