The alternative Dunford-Pettis property on projective tensor products

Dunford–Pettis properties, Hilbert spaces and projective tensor products

“…By [38,Corollary 3.3], Cx ⊗ y is M-orthogonal to each Cx n ⊗ y n and by [10, Lemma 2.1] (or see [31,Lemma 2]) (x n ⊗ y n ) is weakly null. Hence,…”

“…Looking inside G ∼ = C[0, 1], we can choose a norm-one element v in G and a continuous projection R on G such that Y = R(G) ∼ = G, R(v) = 0 and Cv is M-orthogonal to R(G) (compare the proof of [38,Theorem 3.5]). We have a composition of surjective operators…”

“…In [10] it was proved that for infinite compact Hausdorff spaces K 1 and K 2 , that C(K 1 )⊗ π C(K 2 ) has the Dunford-Pettis property if and only if K 1 and K 2 are scattered, a result extended to generalised operator algebras in [7] with a proof that for infinite-dimensional JB * -triples E and F , E⊗ π F has the Dunford-Pettis property if and only if both E and F have the Dunford-Pettis property and do not contain an isomorphic copy of 1 . It was shown in [38] that if E and F are JB * -triples containing an isomorphic copy of c 0 and 1 , respectively, then E⊗ π F does not have the DP1. JB * -triples form the widely studied class of complex Banach spaces whose open unit ball is a bounded symmetric domain.…”

“…Subsequently introduced variations include the surjective Dunford-Pettis property [36] and the more geometric alternative Dunford-Pettis property, known as the DP1, of [28] (see Section 2 for definitions). The question [21] of whether the Dunford-Pettis property is preserved by projective tensor products was negated in [43] and Dunford-Pettis theory of projective tensor products has been widely studied since, in [7,10,26,27,31,38,42], for example. In [10] it was proved that for infinite compact Hausdorff spaces K 1 and K 2 , that C(K 1 )⊗ π C(K 2 ) has the Dunford-Pettis property if and only if K 1 and K 2 are scattered, a result extended to generalised operator algebras in [7] with a proof that for infinite-dimensional JB * -triples E and F , E⊗ π F has the Dunford-Pettis property if and only if both E and F have the Dunford-Pettis property and do not contain an isomorphic copy of 1 .…”

The Alternative Dunford-Pettis Property in Subspaces of Operator Ideals

“…This means that for every weakly convergent sequence x n → x in X with x n = x = 1, we have T x n − T x → 0. We refer the reader to [5], [6] and [15] for valuable results on DP1.…”