Erratum to: Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1≤p≤q Weak* compactness of the closed unit ball of a Köthe dual space

Erratum to: Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1 ≤ p ≤ q Weak* compactness of the closed unit ball of a Köthe dual space O. Delgado1 · E. A. Sánchez Pérez2 Abstract Let X be a saturated Banach function space and denote by X its Köthe dual. In the paper (Delgado and Sánchez Pérez in Positivity 20:999–1014, 2016) referenced in the title it is implicitly used that the closed unit ball BX of X is compact for the topology σ(X, X) on X defined by the elements of X. This fact could be not true in general if X is not σ -order continuous. Keywords Banach function space · Köthe dual · Weak* compactness · σ -Order continuity O. Delgado gratefully acknowledge the support of the Ministerio de Economía y Competitividad (project #MTM2015-65888-C4-1-P) and the Junta de Andalucía (project FQM-7276), Spain. E. A. Sánchez Pérez acknowledges with thanks the support of the Ministerio de Economía y Competitividad (project #MTM2012-36740-C02-02), Spain. BE. A. Sánchez Pérez [email protected].es O. Delgado [email protected] 1Departamento de Matemática Aplicada I, E. T. S. de Ingeniería de Edificación, Universidad de Sevilla, 41012 Sevilla, Spain 2Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 Valencia, Spain Let (Ω,Σ,μ) be a σ-finite measure space and denote by L0(μ) the space of real measurable functions defined on Ω. Consider a saturated Banach function space X, that is, a Banach lattice that is an ideal in L0(μ) such that there is no A∈Σwith μ(A)>0 satisfying that fχA=0μ-a.e. for all f∈X. The space Xis said to be σ-order continuous if for every (fn)⊂Xwith fn↓0 μ-a.e. it follows that fnX↓0. The Köthe dual Xof Xis the space of functions g∈L0(μ) such that |fg|dμ<∞for all f∈X. It is a saturated Banach function space with norm gX=sup f∈BX  fgdμ  ,where BXdenotes the closed unit ball of X. The space Xcan be identified with a closed subspace of the topological dual X∗of Xvia the linear isometry η:X→X∗given by η(g), f=fgdμfor all g∈Xand f∈X.Themapηis surjective if and only if Xis σ-order continuous. The Köthe bidual X of Xis defined as X := (X). Consider now the weak topology σ(X,X)defined by Xon its Köthe dual X. Through all the recently published paper [1] referenced in the title, the fact that the closed unit ball BXof Xis σ(X,X)-compact is used, but this is not in general true. For getting a counterexample we just have to take X=L∞(μ) for which X=L1(μ) and note that, by the Banach–Bourbaki theorem, BL1(μ) is not compact for the topology σ(L1(μ), L∞(μ)). However BXis σ(X,X)-compact whenever Xis σ-order continuous, since in this case BX∗is identified with BXvia the map η−1:X∗→Xand we can apply the Banach-Alaoglu theorem. Therefore, requiring Xto have the natural condition of σ-order continuity all the results in [1] hold. A characterization of the weak* compactness of BXfollows as a particular case of a known result for Riesz spaces, see [2, Theorem 82G and Proposition 82B]. Recall that the σ-order continuous part Xaof Xis defined as the largest σ-order continuous closed solid subspace of Xand can be described as Xa=f∈X:|f|≥ fn↓0 implies fnX↓0. Then we have that BXis σX,X-compact if and only if fnX ↓0 whenever (fn)⊂Xwith fn↓0μ-a.e., that is, if and only if X ⊂(X)a. Finally, note that we can obtain compactness for BXif we consider the weak topology on Xdefined by the elements of the σ-order continuous part Xaof X. Indeed, in the case when Xahas a weak unit we have that Xais super order dense in X and so it can be proved that (Xa)=Xwith equal norms. Then, since Xais σ-order continuous, it follows that BXis σX,Xa-compact. Acknowledgements The authors would like to thank the referee for providing the reference [2]. References 1. Delgado, O., Sánchez Pérez, E.A.: Strong extensions for q-summing operators acting in p-convex Banach function spaces for 1 ≤p≤q. Positivity 20, 999–1014 (2016) 2. Fremlin, D.H.: Topological Riesz Spaces and Measure Theory. Cambridge University Press, Cambridge (1974)