Projective Tensor Product Research Articles

Points of differentiability of the norm in Lipschitz-free spaces

We consider convex series of molecules in Lipschitz-free spaces, i.e. elements of the form μ=∑nλnδxn−δynd(xn,yn) such that ‖μ‖=∑n|λn|. We characterise these elements in terms of geometric conditions on the points xn, yn of the underlying metric space, and determine when they are points of Gâteaux differentiability of the norm. In particular, we show that Gâteaux and Fréchet differentiability are equivalent for finitely supported elements of Lipschitz-free spaces over uniformly discrete and bounded metric spaces, and that their tensor products with Gâteaux (resp. Fréchet) differentiable elements of a Banach space are Gâteaux (resp. Fréchet) differentiable in the corresponding projective tensor product.

Module approximate biprojectivity and module approximate biflatness of Banach algebras

In this paper, in analogy with the classical case, we introduce module approximately biprojective and module approximately biflat Banach algebras. For an inverse semigroup S with the set of idempotents E, we show that the concepts of module approximate biprojectivity and module pseudo-amenability for $$l^1(S,\omega )$$ coincide (as an $$l^1(E)$$ -module). We also find necessary and sufficient conditions for the weighted semigroup algebra $$l^{1}(S,\omega )$$ to be module approximately biprojective and module approximately biflat. Finally, we study the module cohomological properties of projective tensor product of Banach algebras.

Lipschitz p-summing multilinear operators

We apply the geometric approach provided by Σ-operators to develop a theory of p-summability for multilinear operators. In this way, we introduce the notion of Lipschitz p-summing multilinear operators and show that it is consistent with a general panorama of generalization: Namely, they satisfy Pietsch-type domination and factorization theorems and generalizations of the inclusion Theorem, Grothendieck's coincidence Theorems, the weak Dvoretsky-Rogers Theorem and a Lindenstrauss-Pełczyńsky Theorem. We also characterize this new class in tensorial terms by means of a Chevet-Saphar-type tensor norm. Moreover, we introduce the notion of Dunford-Pettis multilinear operators. With them, we characterize when a projective tensor product contains ℓ1. Relations between Lipschitz p-summing multilinear operators with Dunford-Pettis and Hilbert-Schmidt multilinear operators are given.

On Homotopy Invariants of Tensor Products of Banach Algebras

We generalize results of Davie and Raeburn describing homotopy types of the group of invertible elements and of the set of idempotents of the projective tensor product of complex unital Banach algebras. We illustrate our results by specific examples.

Cones of completely bounded maps

By analogy with the Choi matrix we associate an operator $$C_{\varphi }\in \mathrm{B}({\mathscr {H}})$$ to each weak* continuous $${\mathscr {A}}$$ -bimodule map $$\varphi :\mathrm{B}({\mathscr {K}})\rightarrow \mathrm{B}({\mathscr {H}})$$ , where $${\mathscr {K}}$$ and $${\mathscr {H}}$$ are normal Hilbert modules over a von Neumann algebra $${\mathscr {A}}$$ and $${\mathscr {K}}$$ contains a cyclic vector for $${\mathscr {A}}$$ . If $${\mathscr {A}}\subseteq \mathrm{B}({\mathscr {K}})$$ has no central summands of type I ( $${\mathscr {K}}$$ cyclic), every normal $${\mathscr {A}}$$ -bimodule map on $$\mathrm{B}({\mathscr {K}})$$ , which is positive on $${\mathscr {A}}^{\prime }$$ , is shown to be completely positive on $${\mathscr {Z}}^{\prime }$$ , where $${\mathscr {A}}^{\prime }$$ and $${\mathscr {Z}}^{\prime }$$ are the commutant of $${\mathscr {A}}$$ and the center $${\mathscr {Z}}$$ of $${\mathscr {A}}$$ . We investigate cones of bimodule maps, introduce the corresponding dual cones of operators and show that in an appropriate context these notions reduce to those studied earlier by Stormer. We also consider positive maps relative to a mapping cone and positivity in operator projective tensor product of suitable operator bimodules.

Universal Gaps for XOR Games from Estimates on Tensor Norm Ratios

We define and study XOR games in the framework of general probabilistic theories, which encompasses all physical models whose predictive power obeys minimal requirements. The bias of an XOR game under local or global strategies is shown to be given by a certain injective or projective tensor norm, respectively. The intrinsic (i.e. model-independent) advantage of global over local strategies is thus connected to a universal function r(n, m) called ‘projective–injective ratio’. This is defined as the minimal constant rho such that Vert cdot Vert _{Xotimes _pi Y}leqslant rho ,Vert cdot Vert _{Xotimes _varepsilon Y} holds for all Banach spaces of dimensions dim X=n and dim Y=m, where Xotimes _pi Y and X otimes _varepsilon Y are the projective and injective tensor products. By requiring that X=Y, one obtains a symmetrised version of the above ratio, denoted by r_s(n). We prove that r(n,m)geqslant 19/18 for all n,mgeqslant 2, implying that injective and projective tensor products are never isometric. We then study the asymptotic behaviour of r(n, m) and r_s(n), showing that, up to log factors: r_s(n) is of the order sqrt{n} (which is sharp); r(n, n) is at least of the order n^{1/6}; and r(n, m) grows at least as min {n,m}^{1/8}. These results constitute our main contribution to the theory of tensor norms. In our proof, a crucial role is played by an ‘ell _1/ell _2/ell _{infty } trichotomy theorem’ based on ideas by Pisier, Rudelson, Szarek, and Tomczak-Jaegermann. The main operational consequence we draw is that there is a universal gap between local and global strategies in general XOR games, and that this grows as a power of the minimal local dimension. In the quantum case, we are able to determine this gap up to universal constants. As a corollary, we obtain an improved bound on the scaling of the maximal quantum data hiding efficiency against local measurements.

The Fremlin projective tensor product of Banach lattice algebras

In this paper we prove that the Fremlin projective tensor product of Banach lattice algebras can be endowed with a Banach lattice algebra structure and satisfies an appropriate universal property. In addition, we prove that the Fremlin projective tensor product of Banach f-algebras is a Banach f-algebra.

Automorphisms of the Banach space projective tensor product of $$C^*$$-algebras

For unital $$C^*$$-algebras A and B, we completely characterize the isometric ($$*$$-) automorphisms of their Banach space projective tensor product $$A\otimes ^\gamma B$$. This leads to the characterization of inner and outer isometric $$*$$-automorphisms of $$A\otimes ^\gamma B$$ as well. As an application, we provide a partial affirmative answer to a question posed by Kaijser and Sinclair, viz., we prove that for unital $$C^*$$-algebras A and B, the set of norm-one unitaries of $$A\otimes ^\gamma B$$ coincides with $$U(A) \otimes U(B)$$, where U(A) is the unitary group of A. We also establish the fact that the relative commutant of $$A\otimes ^\gamma {\mathbb {C}}1$$ in $$A \otimes ^\gamma B$$ is the same as $${\mathcal {Z}}(A) \otimes ^\gamma B$$, where B is a subhomogenous unital $$C^*$$-algebra, and A is any $$C^*$$-algebra.

Almost squareness and strong diameter two property in tensor product spaces

We study almost squareness and the strong diameter two property in the setting of projective (symmetric) tensor product of Banach spaces. We prove that almost squareness is stable by taking projective tensor products, providing non-trivial examples of ASQ projective tensor products of Banach spaces. Furthermore, we give sufficient conditions for a projective symmetric tensor product to have the strong diameter two property. This extend most of the previously known results and provide new examples of projective symmetric tensor product spaces with the strong diameter two property.

On Kalton’s theorem for regular compact operators and Grothendieck property for positive projective tensor products

On Kalton’s theorem for regular compact operators and Grothendieck property for positive projective tensor products

Strong subdifferentiability and local Bishop–Phelps–Bollobás properties

Some local versions of the Bishop–Phelps–Bollobás property for operators have been recently presented in Dantas et al. (J Math Anal Appl 468(1):304–323, 2018). In the present article, we continue studying these properties for multilinear mappings. We show some differences between the local and uniform versions of these and also provide some interesting examples which show that this study is not just a mere generalization of the linear case. As a consequence of our results, we get that, for \(2<p,q<\infty \), the norm of the projective tensor product \(\ell _p \hat{\otimes }_{\pi } \ell _{q}\) is strongly subdifferentiable. Moreover, we present necessary and sufficient conditions for the norm of a Banach space Y to be strongly subdifferentiable through the study of these properties for bilinear mappings on \(\ell _1^N \times Y\).

On Banach space projective tensor product of $$C^*$$-algebras

For \(C^*\)-algebras A and B, we show that there is a natural homeomorphism from the product of spaces of maximal (resp., maximal modular) ideals of A and B onto the space of maximal (resp., maximal modular) ideals of \(A \otimes ^\gamma B\), with respect to the hull-kernel topology. This is based on and preceded by an analysis of the structure of closed ideals of \(A \otimes ^\gamma B\) in terms of those of A and B. During the process, we also identify the center of \(A \otimes ^\gamma B\) with \({\mathcal {Z}}(A) \otimes ^\gamma {\mathcal {Z}}(B)\).

Complementation in the Fremlin vector lattice symmetric tensor products-II

For a vector lattice E and $$n \in \mathbb {N}$$, let $${\bar{\otimes }}_{n,s}E$$ denote the n-fold Fremlin vector lattice symmetric tensor product of E. For $$m, n \in \mathbb {N}$$ with $$m > n$$, we prove that (i) if $${\bar{\otimes }}_{m,s}E$$ is uniformly complete then $${\bar{\otimes }}_{n,s}E$$ is positively isomorphic to a complemented subspace of $${\bar{\otimes }}_{m,s}E$$, and (ii) if there exists such that $$\ker (\phi )$$ is a projection band in E then $${\bar{\otimes }}_{n,s}E$$ is lattice isomorphic to a projection band of $${\bar{\otimes }}_{m,s}E$$. We also obtain analogous results for the n-fold Fremlin projective symmetric tensor product $${\hat{\otimes }}_{n,s,|\pi |}E$$ of E where E is a Banach lattice.

A study of reciprocal Dunford–Pettis-like properties on Banach spaces

In this article, we study the relationship between p- (V) subsets and p-$$ (V^{*})$$ subsets of dual spaces. We investigate the Banach space X with the property that the adjoint of every p-convergent operator $$ T:X\rightarrow Y $$ is weakly q-compact, for every Banach space Y. Moreover, we define the notion of q-reciprocal Dunford–Pettis$$ ^{*} $$ property of order p on Banach spaces and obtain a characterization of Banach spaces with this property. Also, the stability of reciprocal Dunford–Pettis property of order p for projective tensor product is given.

Geometry of C⁎-algebras, and the bidual of their projective tensor product

Given C⁎-algebras A and B, consider the Banach algebra A⊗γB, where ⊗γ denotes the projective Banach space tensor product. If A and B are commutative, this is the Varopoulos algebra VA,B; we write VA for VA,A. It has been an open problem for almost 40 years to determine precisely when A⊗γB is Arens regular; see, e.g., [33], [48], [49]. We solve this classical question for arbitrary C⁎-algebras. Indeed, we show that A⊗γB is Arens regular if and only if A or B has the Phillips property; note that A has the latter property if and only if it is scattered and has the Dunford–Pettis Property. A further equivalent condition is that A⁎ has the Schur property, or, again equivalently, the enveloping von Neumann algebra A⁎⁎ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A⊗γB is entirely encoded in the geometry of the C⁎-algebras. In case A and B are von Neumann algebra, we conclude that A⊗γB is Arens regular (if and) only if A or B is finite-dimensional. We also show that this characterization does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C⁎-algebras A and B, we obtain that VA,B is Arens regular if and only if A or B is scattered. We further describe the centre Z(VA⁎⁎), showing that it is Banach algebra isomorphic to A⁎⁎⊗ehA⁎⁎, where ⊗eh denotes the extended Haagerup tensor product. We deduce that VA is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, VA is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered; this is the case, e.g., for A=ℓ∞.

The reciprocal Dunford--Pettis property of order $p$ in projective tensor products

We investigate whether the projective tensor product of two Banach spaces $X$ and $Y$ has the reciprocal Dunford--Pettis property of order $p$, $1\le\allowbreak p<\infty$, when $X$ and $Y$ have the respective property.

On closed Lie ideals of certain tensor products of C∗‐algebras II

AbstractWe identify all closed Lie ideals of and , where is either the Haagerup tensor product, the Banach space projective tensor product or the operator space projective tensor product, A is any simple ‐algebra, B is any ‐algebra with one of them admitting no tracial states, and H is an infinite dimensional separable Hilbert space. Further, generalizing a result of Marcoux, we also identify all closed Lie ideals of , where A is a simple ‐algebra with at most one tracial state and B is any commutative ‐algebra.

DAUGAVET PROPERTY IN TENSOR PRODUCT SPACES

AbstractWe study the Daugavet property in tensor products of Banach spaces. We show that$L_{1}(\unicode[STIX]{x1D707})\widehat{\otimes }_{\unicode[STIX]{x1D700}}L_{1}(\unicode[STIX]{x1D708})$has the Daugavet property when$\unicode[STIX]{x1D707}$and$\unicode[STIX]{x1D708}$are purely non-atomic measures. Also, we show that$X\widehat{\otimes }_{\unicode[STIX]{x1D70B}}Y$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 techniques, we also obtain consequences about roughness in projective tensor products as well as the Daugavet property of projective symmetric tensor products.

Johnson pseudo-contractibility of second duals of certain Banach algebras

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is a finite amenable group, where [Formula: see text] is an archimedean semigroup. We also show that the matrix algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

On the structure of the set of positive maps

The full description of the set of positive maps $$T: {\mathfrak {A}}\rightarrow {\mathcal {B}}({\mathcal {H}})$$ ( $${\mathfrak {A}}$$ a $$C^*$$ -algebra) is given. The approach is based on the simple prescription for selecting various types of positive maps. This prescription stems from the Grothendieck theory of projective tensor products complemented by the theory of tensor connes. In particular, the origin of non-decomposable maps is clarified.