Some results on almost square Banach spaces

Symmetric Strong Diameter Two Property

“…However, we do not know whether every space with the SSD2P contains c 0 . On the other hand, every Banach space containing a copy of c 0 can be equivalently renormed to have the SSD2P, in fact even to be ASQ (see [8]). Let us summarize the results of the paper.…”

Unconditional almost squareness and applications to spaces of Lipschitz functions

“…Note that this provides a partial answer to the problem of whether there exists any dual ASQ Banach space posed in [1] and [3]. Above Theorem says that the answer is no if we additionally assume unconditional almost squareness.…”

“…Roughly speaking, we can say that ASQ Banach spaces have a strong c 0 behaviour from a geometrical point of view. This c 0 behaviour is encoded by the fact that a Banach space admits an equivalent renorming to be ASQ if, and only if, the space contains an isomorphic copy of c 0 [3]. In this setting, it could be natural wondering whether this kind of spaces can be dual ones.…”

Almost square and octahedral norms in tensor products of Banach spaces

“…The dual, ( ⊗ π,s,N X) * = P( N X), is the Banach space of N-homogeneous continuous polynomials on X (see [18] for background). It is known that ⊗ π,s,N X has the SD2P provided the Banach space X is ASQ [13,Theorem 3.3]. The proof of this result relies heavily on the fact that the sequences involved in the definition of ASQ can be chosen to be c 0 -sequences (see [3,Lemma 2.6]).…”

“…However, the assumption of having infinite-dimensional centralizer have been shown to be far from necessary. In the symmetric case it has been recently proved that the symmetric projective tensor product of any ASQ space has the SD2P [13,Theorem 3.3]. Furthermore, in the non-symmetric case there are even stability results for some diameter two properties, e.g.…”