Related papers: Almost square and octahedral norms in tensor produ…

We study almost square Banach spaces under a topological point of view. Indeed, we prove that the class of Banach spaces which admits an equivalent norm to be ASQ is that of those Banach spaces which contain an isomorphic copy of $c_0$. We…

We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. Firstly, we will prove the existence of a Banach space $Y$ such that the injective tensor products $l_1\widehat{\otimes}_\varepsilon Y$…

We continue the investigation of the behaviour of diameter two properties in tensor products of Banach spaces. Our main result shows that the symmetric strong diameter two property is stable by taking projective tensor products. We also…

We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product. We prove that almost squareness is preserved by taking projective tensor product, providing non-trivial examples of ASQ…

We prove that there exists a finite-dimensional Banach space $X$ such that $L_1^\mathbb C([0,1])\widehat{\otimes}_\varepsilon X$ fails the strong diameter two property and $L_\infty^\mathbb C([0,1])\widehat{\otimes}_\pi X^*$ fails to have…

We single out and study a natural class of Banach spaces -- almost square Banach spaces. In an almost square space we can find, given a finite set $x_1,x_2,\ldots,x_N$ in the unit sphere, a unit vector $y$ such that $\|x_i-y\|$ is almost…

We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.

We study octahedral norms in the space of bounded linear operators between Banach spaces. In fact, we prove that $L(X,Y)$ has octahedral norm whenever $X^*$ and $Y$ have octahedral norm. As a consequence the space of operators $L(\ell_1…

For $C^{*}$-algebras $A$ and $B$, the operator space projective tensor product $A\hat{\otimes}B$ and the Banach space projective tensor product $A\otimes_{\gamma}B$ are shown to be symmetric. We also show that $A\hat{\otimes}B$ is weakly…

A Banach space is locally almost square if, for every $y$ in its unit sphere, there exists a sequence $(x_n)$ in its unit sphere such that $\lim\|y\pm x_n\|=1$. A Banach space is weakly almost square if, in addition, we require the sequence…

Quasi *-algebras form an essential class of partial *-algebras, which are algebras of unbounded operators. In this work, we aim to construct tensor products of normed, respectively Banach quasi *-algebras, and study their capacity to…

We show that finite dimensional Banach spaces fail to be uniformly non locally almost square. Moreover, we construct an equivalent almost square bidual norm on $\ell_\infty.$ As a consequence we get that every dual Banach space containing…

It is known that a Banach space contains an isomorphic copy of $c_0$ if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate…

We prove some results on weakly almost square Banach spaces and their relatives. On the one hand, we discuss weak almost squareness in the setting of Banach function spaces. More precisely, let $(\Omega,\Sigma)$ be a measurable space, let…

We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of…

We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem,…

Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.

We show that for a Banach algebra $A$ with a bounded approximate identity, the amenability of the projective tensor product of A with A, the amenability of the projective tensor product of A with A^{op}and the amenability of A are…

Let $X$ be a complex Banach space and $x,y\in X$. By definition, we say that $x$ is Birkhoff-James orthogonal to $y$ if $ \|x+\lambda y\|_{X} \geq \|x\|_{X}$ for all $\lambda \in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal…

It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and…