Let $m,n$ be positive integers with $m<n$. Under certain assumptions on the Banach space $X$, we prove that the $n$-fold projective tensor product of $X$, $\widehat{\otimes}^n_πX$, is not isomorphic to any subspace of any quotient of the $m$-fold projective tensor product, $\widehat{\otimes}_π^m X$. In particular, we prove that $\widehat{\otimes}^n_πc_0$ is not isomorphic to any subspace of any quotient of $\widehat{\otimes}_π^m c_0$.
A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a `matrix-free' language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to completely bounded bilinear operators. We study some general properties of this tensor product (among them a kind of adjoint associativity), and compute it for some tensor factors, notably for $L_1$ spaces. In particular, we obtain what could be called the proto-quantum version of the Grothendieck theorem about classical projective tensor product…
We introduce lexicographic cones, a method of assigning an ordered vector space $\Lex(S)$ to a poset $S$, generalising the standard lexicographic cone. These lexicographic cones are then used to prove that the projective tensor cone of two arbitrary cones is a cone, and to find a new characterisation of finite-dimensional vector lattices.
We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1<p,q<\infty$ be such that $1/p+1/q\geq 1$. Let $X$ (resp., $Y$) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower $p$-estimate (resp., $q$-estimate). If $X$ and $Y$ are strongly weakly compactly generated, then so is its projective tensor product $X \widehat{\otimes}_πY$.
We characterize the positive Schur property in the positive projective tensor products of Banach lattices, we establish the connection with the weak operator topology and we give necessary and sufficient conditions for the space of regular multilinear operators/homogeneous polynomials taking values in a Dedekind complete Banach lattice to have the positive Schur property.
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 projective tensor product spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property which extend most of the known results and provide new examples of such spaces with the strong diameter two property.
Czech Technical University in Prague（布拉格捷克理工大学）；Instituto Balseiro, CNEA – Universidad Nacional de Cuyo, CONICET（巴尔塞罗研究所，国家原子能委员会–门多萨国立大学，阿根廷国家科学研究委员会）
Galego and Samuel showed that if $K,L$ are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_πC(L)$ is $c_0$-saturated if and only if it is subprojective if and only if $K$ and $L$ are both scattered. We remove the hypothesis of metrizability from their result, and extend it from the case of the two-fold projective tensor product to the general $n$-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces $K_1, \ldots, K_n$, $\widehat{\otimes}_{π, i=1}^n C(K_i)$ is $c_0$-saturated if and only if it is subprojective if and only if each $…
The tensor product of two ordered vector spaces can be ordered in more than one way, just as the tensor product of normed spaces can be normed in multiple ways. Two natural orderings have received considerable attention in the past, namely the ones given by the projective and injective (or biprojective) cones. This paper aims to show that these two cones behave similarly to their normed counterparts, and furthermore extends the study of these two cones from the algebraic tensor product to completed locally convex tensor products. The main results in this paper are the following: (i) drawing pa…
In this paper, we are interested in studying when an element $z$ in the projective tensor product $X \widehat{\otimes}_πY$ turns out to be a Daugavet point. We prove first that, under some hypothesis, the assumption of $X \widehat{\otimes}_πY$ having the Daugavet property implies the existence of a great amount of isometries from $Y$ into $X^*$. Having this in mind, we provide methods for constructing non-trivial Daugavet points in $X \widehat{\otimes}_πY$. We show that $C(K)$-spaces are examples of Banach spaces such that the set of the Daugavet points in $C(K) \widehat{\otimes}_πY$ is weakly…
In this work, we study the volume ratio of the projective tensor products $\ell^n_p\otimes_π\ell_q^n\otimes_π\ell_r^n$ with $1\leq p\leq q \leq r \leq \infty$. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kashin type whenever $1\leq p \leq q\leq r \leq 2$ or $1\leq p \leq 2 \leq r \leq \infty$ and $q=2$. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype $2$ constant, we obtain information on the cotype of these $3$-fold projective t…
Let $(E,F)$ be a pair of Fréchet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_π F$. Specifically we focus on the two properties generalizing Köthe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fréchet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in…
We show that all the symmetric projective tensor products of a Banach space $X$ have the Daugavet property provided $X$ has the Daugavet property and either $X$ is an $L_1$-predual (i.e.\ $X^*$ is isometric to an $L_1$-space) or $X$ is a vector-valued $L_1$-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise "localised" versions of the Daugavet property (i.e.\ Daugavet points and $Δ$-points recently introduced) for $L_1$-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a pol…
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 equivalent. Also if A is a closed ideal in a commutative Banach algebra B, then the weak amenability of the projective tensor product of A and B implies the weak amenability of A.
As is known, there exists an alternative, "non-matricial" way to present basic notions and results of quantum functional analysis (= operator space theory). This approach is based on considering, instead of matrix spaces, a single space, consisting, roughly speaking, of vectors from the initial linear space equipped with coefficients taken from some good operator algebra. It seems that so far there was no systematical exposition of the theory in the framework of the non-matricial approach. We believe, however, that in a number of topics the non-matricial approach gives a more elegant and trans…
The tensor product of the Fremlin tensor product of Banach lattice algebras
Research Laboratory of Algebra, Topology, Arithmetic, and Order（代数、拓扑、算术与序研究实验室）；Department of Mathematics（数学系）；Faculty of Mathematical, Physical and Natural Sciences of Tunis（突尼斯数学、物理与自然科学学院）；Tunis-El Manar University（突尼斯埃尔马纳尔大学）
We prove a general result on the factorization of matrix-valued analytic functions. We deduce that if $(E_0,E_1)$ and $(F_0,F_1)$ are interpolation pairs with dense intersections, then under some conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, we have $$ [E_0\hat\otimes F_0,E_1\hat\otimes F_1]_t= [E_0, E_1]_t\hat\otimes[F_0, F_1]_t, 0 < t< 1.$$ We find also conditions on the spaces $E_0$, $E_1$, $F_0$ and $F_1$, so that the following holds $$ [E_0\wcheck\otimes F_0,E_1\wcheck\otimes F_1]_t= [E_0,E_1]_t\wcheck\otimes [F_0,F_1]_t, 0 <t< 1.$$ Some applications of these results are also co…
It is shown that for every $p\in (1,\infty)$ there exists a Banach space $X$ of finite cotype such that the projective tensor product $\ell_p\tp X$ fails to have finite cotype. More generally, if $p_1,p_2,p_3\in (1,\infty)$ satisfy $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\le 1$ then $\ell_{p_1}\tp\ell_{p_2}\tp\ell_{p_3}$ does not have finite cotype. This is a proved via a connection to the theory of locally decodable codes.
We study the Daugavet property in tensor products of Banach spaces. We show that $L_1(μ)\widehat{\otimes}_\varepsilon L_1(ν)$ has the Daugavet property when $μ$ and $ν$ are purely non-atomic measures. Also, we show that $X\widehat{\otimes}_πY$ has the Daugavet property provided $X$ and $Y$ are $L_1$-preduals with the Daugavet property, in particular spaces of continuous functions with this property. With the same tecniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.
We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.
In this paper we prove that the $3$-fold injective tensor product $\ell_1 \widehat{\otimes}_\varepsilon \ell_1 \widehat{\otimes}_\varepsilon \ell_1 $ is not isomorphic to any subspace of $\ell_1 \widehat{\otimes}_\varepsilon \ell_1$. This result provides a new solution to a problem of Diestel on the projective tensor products of $c_0.$ Moreover, this result implies that for any infinite countable compact space $K,$ the $3$-fold projective tensor product $C(K) \widehat{\otimes}_πC(K)\widehat{\otimes}_πC(K)$ is not isomorphic to any quotient of $C(K) \widehat{\otimes}_πC(K)$.
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+λy\|_{X} \geq \|x\|_{X}$ for all $λ\in \mathbb{C}$. We prove that $x$ is Birkhoff-James orthogonal to $y$ if and only if there exists a semi-inner product $φ$ on $X$ such that $\|φ\| = 1$, $φ(x,x)=\|x\|^2$ and $φ(x,y)=0$. A similar result holds for $C^*$-algebras. A key point in our approach to orthogonality is the representations of bounded bilinear maps via projective tensor product spaces.
Tensor products of convex cones have recently come up in different areas, ranging from functional analysis and operator theory to approximation theory and theoretical physics. However, most of the existing literature focuses either on Archimedean lattice cones or on closed proper cones in finite-dimensional spaces, thereby excluding many cones, including even standard cones such as an infinite-dimensional positive semidefinite cone. For general cones, results are few and far between, and many basic questions remain unanswered. In this memoir, we develop the theory of tensor products of convex …
We prove that, given two Banach spaces $X$ and $Y$ and bounded, closed convex sets $C\subseteq X$ and $D\subseteq Y$, if a nonzero element $z\in \overline{\mathrm{co}}(C\otimes D)\subseteq X\widehat{\otimes}_πY$ is a preserved extreme point then $z=x_0\otimes y_0$ for some preserved extreme points $x_0\in C$ and $y_0\in D$, whenever $K(X,Y^*)$ separates points of $X \widehat{\otimes}_πY$ (in particular, whenever $X$ or $Y$ has the compact approximation property). Moreover, we prove that if $x_0\in C$ and $y_0\in D$ are weak-strongly exposed points then $x_0\otimes y_0$ is weak-strongly exposed…
The injective tensor product of normal representable bimodules over von Neumann algebras is shown to be normal. The usual Banach module projective tensor product of central representable bimodules over an Abelian C$^*$-algebra is shown to be representable. A normal version of the projective tensor product is introduced for central normal bimodules.
We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $S_1$ to the Euclidean space $\ell_2$, which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that pr…
We study the ranges of the maps of convolution $u\otimes v\mapsto u\ast v$ and a `twisted' convolution $u\otimes v\mapsto u\ast \check{v}$ ($\check{u}(s)=u(s^{-1})$) and on the Haagerup tensor product of a Fourier algebra of a compact group $A(G)$ with itself. We compare the results to result of factoring these maps through projective and operator projective tensor products. We notice that $(A(G),\ast)$ is an operator algebra and observe an unexpected set of spectral synthesis.
We show analogues of the Daniell-Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on $T_0$ topological spaces.
Let $\mathcal{A}$ be a $C^*$-algebra, and consider the Banach algebra $\mathcal{A} \otimes_γ\mathcal{A}$, where $\otimes_γ$ denotes the projective Banach space tensor product; if $\mathcal{A}$ is commutative, this is the Varopoulos algebra $V_\mathcal{A}$. It has been an open problem for more than 35 years to determine precisely when $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular. Even the situation for commutative $\mathcal{A}$, in particular the case $\mathcal{A} = \ell_\infty$, has remained unsolved. We solve this classical question for arbitrary $C^*$-algebras by using von Neumann alg…
