Locally quasi-nilpotent elementary operators

Let A be a unital dense algebra of linear mappings on a complex vector space X. Let φ = Σⁿ _i=1 M_ai,_bi be a locally quasi-nilpotent elementary operator of length n on A. We show that, if {a1, . . . , an} is locally linearly independent, then the local dimension of V (φ) = span{b_ia_j : 1 ≤ i, j ≤ n} is at most n(n−1) 2 . If ldim V (φ) = n(n−1) 2 , then there exists a representation of φ as φ = Σⁿ _i=1 M_ui,_vi with v_iu_j = 0 for i ≥ j. Moreover, we give a complete characterization of locally quasinilpotent elementary operators of length 3.

On the positive approximation property

Using a different approach to that of Popa, we arrive at an alternative definition
of the positive approximation property for order complete Banach lattices.
Some results associated with this new approach may be of independent interest. We
also prove a Banach lattice analogue of an old characterization, due to Palmer, of
the metric approximation property in terms of the continuous bidual of the ideal of
approximable operators.