Symmetric Tensor Product Research Articles

Symmetric and antisymmetric tensor products for the function-theoretic operator theorist

Abstract We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

A version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices

Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${\mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $\hat{\otimes}_{n,s,|\pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.

Daugavet property in projective symmetric tensor products of Banach spaces

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 Delta-points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for L_1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L_1-preduals in terms of the absence of Delta-points and also to provide new examples of L_1-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L_1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.

On norm-attainment in (symmetric) tensor products

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product of a Banach space X, which turns out to be naturally related to the classical norm-attainment of N -homogeneous polynomials on X. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in is dense. We show that the set of all normattaining symmetric tensors is dense in for a large set of Banach spaces such as Lp -spaces, isometric L 1-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if X* satisfies the Radon-Nikodým and approximation properties, then the set of all norm-attaining symmetric tensors in is dense. From these techniques, we can present new examples of Banach spaces X and Y such that the set of all norm-attaining tensors in the projective tensor product is dense, answering positively an open question from the paper [10].

Twistorial description of Bondi-Metzner-Sachs symmetries at null infinity

We describe a novel twistorial construction of the asymptotic BMS symmetries at null infinity for asymptotically flat spacetimes. We define BMS twistors as spinor solutions to some set of components of the usual spacetime twistor equation restricted to null infinity. The space of BMS twistors is infinite-dimensional. We show that given two BMS twistors their symmetric tensor product can be used to generate (complex) vector fields which are the infinitesimal BMS symmetries of null infinity. In this sense BMS twistors are "square roots" of BMS symmetries. We also show that these BMS twistor equations can be written a pair of covariant spinor-valued equations which are completely determined by the intrinsic universal structure of null infinity.

The hyper Zagreb index of some product graphs

In this paper some basic mathematical expressions for the Hyper Zagreb index of Product Graphs containing the symmetric difference, disjunction and tensor product have been computed.

More on Wilson toroidal networks and torus blocks

We consider the Wilson line networks of the Chern-Simons 3d gravity theory with toroidal boundary conditions which calculate global conformal blocks of degenerate quasi-primary operators in torus 2d CFT. After general discussion that summarizes and further extends results known in the literature we explicitly obtain the one-point torus block and two-point torus blocks through particular matrix elements of toroidal Wilson network operators in irreducible finite-dimensional representations of sl(2, ℝ) algebra. The resulting expressions are given in two alternative forms using different ways to treat multiple tensor products of sl(2, ℝ) representations: (1) 3mj Wigner symbols and intertwiners of higher valence, (2) totally symmetric tensor products of the fundamental sl(2, ℝ) representation.

Algebraic and differential properties of polynomial Fourier transformation

Methods of integral transformations of (generalized) functions are widely used in the solution of initial and boundary value problems for partial differential equations. However, many problems in applied mathematics require a nonlinear generalization of distribution spaces. Besides, an algebraic structure of a space of distributions is desirable, which is needed, for example, in quantum field theory.In the article, we use the adjoint operator method as well as technique of symmetric tensor products to extended the Fourier transformation onto the spaces of so-called polynomial rapidly decreasing test functions and polynomial tempered distributions. In such spaces it is possible to solve some Cauchy problems, for example, infinite dimensional heat equation associated with the Gross Laplacian.Algebraic and differential properties of the polynomial Fourier transformation are investigated. We prove some analogical to classical properties of this map. Unlike to the classic case, the spaces of polynomial test and generalized functions have algebraic structure. We prove that polynomial Fourier transformation acts as homomorphism of appropriate algebras. It is clear that the classical analogue of such property is absent.

The Drinfeld centre of a symmetric fusion category is 2-fold monoidal

We show that the Drinfeld centre of a symmetric fusion category over an algebraically closed field of characteristic zero is a bilax 2-fold monoidal category. That is, it carries two monoidal structures, the convolution and symmetric tensor products, that are bilax monoidal functors with respect to each other. We additionally show that the braiding and symmetry for the convolution and symmetric tensor products are compatible with this bilax structure.We establish these properties without referring to Tannaka duality for the symmetric fusion category. This has the advantage that all constructions are done purely in terms of the fusion category structure, making the result easy to use in other contexts.

The symmetric tensor product on the Drinfeld centre of a symmetric fusion category

We define a symmetric tensor product on the Drinfeld centre of a symmetric fusion category, in addition to its usual tensor product. We examine what this tensor product looks like under Tannaka duality, identifying the symmetric fusion category with the representation category of a finite (super)-group. Under this identification, the Drinfeld centre is the category of equivariant vector bundles over the finite group (underlying the super-group, in the super case). In the non-super case, we show that the symmetric tensor product corresponds to the fibrewise tensor product of these vector bundles. In the super case, we define for each super-group structure on the finite group a super-version of the fibrewise tensor product. We show that the symmetric tensor product on the Drinfeld centre of the representation category of the resulting finite super-groups corresponds to this super-version of the fibrewise tensor product on the category of equivariant vector bundles over the finite group.

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.

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.

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.

Vacuum initial data on from Killing vectors

We construct compact initial data of constant mean curvature for Einstein’s 4d vacuum equations with positive, where is the cosmological constant, via the conformal method. To construct a transverse, trace-free (TT) momentum tensor explicitly we first observe that, if the seed manifold has two orthogonal Killing vectors, their symmetrized tensor product is a natural TT candidate. Without the orthogonality requirement, but on locally conformally flat seed manifolds there is a generalized construction for the momentum which also involves the derivatives of the Killing fields found in work by Beig and Krammer (2004 Class. Quantum Grav. 21 73). We consider in particular the round three sphere and classify the TT tensors resulting from all possible pairs of its six Killing vectors, focusing on the commuting case where the seed data are —symmetric. As to solving the Lichnerowicz equation, we discuss in particular potential ‘symmetry breaking’ by which we mean that solutions have less symmetries than the equation itself; we compare with the case of the ‘round donut’ of topology . In the absence of symmetry breaking, the Lichnerowicz equation for a symmetric momentum on reduces to an ODE. We analyze distinguished families of solutions and the resulting data via a combination of analytical and numerical techniques. Finally we investigate marginally trapped surfaces of toroidal topology in our data.

A worldline approach to colored particles

Relativistic particle actions are a useful tool to describe quantum field theory effective actions using a string-inspired first-quantized approach. Here we describe how to employ suitable particle actions in the computation of the scalar contribution to the one-loop gluon effective action. We use the well-known method of introducing auxiliary variables that create the color degrees of freedom. In a path integral they implement automatically the path ordering needed to ensure gauge invariance. It is known that the color degrees of freedom introduced this way form a reducible representation of the gauge group. We describe a method of projecting onto the fundamental representation (or any other chosen irrep, if desired) of the gauge group. Previously, we have discussed the case of anticommuting auxiliary variables. Choosing them to be in the fundamental representation allows to obtain, without any extra effort, also the situation in which the color is given by any antisymmetric tensor product of the fundamental. Here, we describe the novel case of bosonic auxiliary variables. They can be used equivalently for creating the color charges in the fundamental representation. In addition one gets, as a byproduct, the cases where the particle can have the color sitting in any symmetric tensor product of the fundamental. This is obtained by tuning to a different value a Chern Simons coupling, present in the model, which controls how the projection is achieved.

Complementation in the Fremlin vector lattice symmetric tensor products-I

For a positive integer n and a uniformly complete vector lattice E, let denote the n-fold Fremlin vector lattice symmetric tensor product of E. We prove that if there exists a lattice homomorphism φ ∈ E ∼ then E is lattice isomorphic to a complemented sublattice of . Moreover, if ker(φ) is a projection band in E then the image of E is also a projection band in .

Symmetric multilinear forms on Hilbert spaces: Where do they attain their norm?

We characterize the sets of norm one vectors x1,…,xk in a Hilbert space H such that there exists a k-linear symmetric form attaining its norm at (x1,…,xk). We prove that in the bilinear case, any two vectors satisfy this property. However, for k≥3 only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to x1,…,xk spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric tensor products and the exposed points of the unit ball of Ls(Hk).

Factorization of (q,p)-summing polynomials through Lorentz spaces

We present a vector valued duality between factorable (q,p)-summing polynomials and (q,p)-summing linear operators on symmetric tensor products of Banach spaces. Several applications are provided. First, we prove a polynomial characterization of cotype of Banach spaces. We also give a variant of Pisier's factorization through Lorentz spaces of factorable (q,p)-summing polynomials from C(K)-spaces. Finally, we show a coincidence result for (q,p)-concave polynomials.

Almost square and octahedral norms in tensor products of Banach spaces

The aim of this note is to study some geometrical properties like diameter two properties, octahedrality and almost squareness in the setting of (symmetric) tensor product spaces. In particular, we show that the injective tensor product of two octahedral Banach spaces is always octahedral, the injective tensor product of an almost square Banach space with any Banach space is almost square, and the injective symmetric tensor product of an octahedral Banach space is octahedral.

Some results on almost square Banach spaces

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 c0. We also prove that the symmetric projective tensor products of an almost square Banach space have the strong diameter two property.