Compact operators in Banach lattices

Completely positive compact operators on non-commutative symmetric spaces

“…It is worth remarking that the preceding proposition shows that the commutative specialisation of Neuhardt's theorem actually coincides with an important special case of one of the main results of [9].…”

“…In addition, a completely positive mapping from a semifinite von Neumann algebra M to any non-commutative space F(σ ) with order continuous norm can be expressed uniquely as the sum of a completely positive compact operator and a completely positive operator which dominates no non-zero compact mapping, in the sense of complete positivity. This is shown in [16] in the case that F is an L p -space, 1 ≤ p < ∞ and in the Banach lattice setting again goes back to [9]. A similar decomposition holds for Dunford-Pettis operators in the case of finite von Neumann algebras.…”

“…In the case of abstract L-spaces this was first proved in [9]. In fact, this result continues to hold for the non-commutative counterparts of separable Lorentz spaces and certain Orlicz spaces.…”

“…In this setting, key technical difficulties arise from the fact that there are no non-commutative counterparts to the arguments of [9] based on properties of order-bounded operators between Banach lattices and technical lattice arguments characterising approximately order-bounded sets in Banach lattices. The key new idea introduced by Neuhardt was to consider domination in the sense of complete positivity, a notion which goes back to Stinespring [18], and to replace arguments in the Banach lattice case based on formulae for the infimum of positive linear operators between Banach lattices by representation theorems for C * -algebras and linear functionals, using the assumption of complete positivity.…”

Incompleteness of the linear span of the positive compact operators

“…0 ≤ S ≤ T ∈ K(E, F ) ⇒S ∈ K(E, F ), see [8] and [15]) and L r (E, F ) is a lattice (or even if it has the Riesz separation property, see [14]) then K r (E, F ) is closed in L r (E, F ) under the regular norm. In particular our counterexample cannot satisfy the Dodds-Fremlin conditions that both E * and F have order continuous norms.…”