scholarly journals Tensor products of l2-valued measures

1976 ◽  
Vol 21 (2) ◽  
pp. 241-246 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Tensor Products ◽  
The Other ◽  
Product Topology ◽  
Arbitrary Vector ◽  
Vector Measures ◽  
Projective Tensor

Duchon (1967, 1969) and Duchon and Kluvánek (1967) considered the problems of the existence of countably additive tensor products for vector measures. Duchon and Kluvánek (1967) showed that a countably additive product with respect to the inductive tensor topology always exists while Kluvánek (1970) presented an example which showed that this was not the case for the projective tensor topology. Kluvánek considered yet another tensor product topology which is stronger than the inductive topology but weaker than the projective tensor topology and showed that a countably additive product for two vector measures always exists for this particular tensor topology, see Kluvánek (1973). He has conjectured that this topology is the strongest tensor topology (given by a cross norm) which always admits products for any two arbitrary vector measures. In this note we use an example of Kluvánek (1974) to show that this conjecture is indeed true when one of the factors in the tensor product is l2 and the other factor is metrizable. The construction used also clarifies a conjecture made by Swartz (to appear) concerning products of Hilbert space valued measures.

scholarly journals SVD on Intersection Spaces

2018 ◽  
Author(s):  
Mazen Ali ◽  
Anthony Nouy
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Tensor Product ◽  
Scalar Product ◽  
Truncation Error ◽  
Tensor Products ◽  
The Other ◽  
Low Rank ◽  
Differentiable Functions ◽  
Low Rank Approximations

We are interested in applying SVD to more general spaces, the motivating example being the Sobolev space $H^1(\Omega)$ of weakly differentiable functions over a domain $\Omega\subset\R^d$. Controlling the truncation error in the energy norm is particularly interesting for PDE applications. To this end, one can apply SVD to tensor products in $H^1(\Omega_1)\otimes H^1(\Omega_2)$ with the induced tensor scalar product. However, the resulting space is not $H^1(\Omega_1\times\Omega_2)$ but is instead the space $H^1_{\text{mix}}(\Omega_1\times\Omega_2)$ of functions with mixed regularity. For large $d>2$ this poses a restrictive regularity requirement on $u\in H^1(\Omega)$. On the other hand, the space $H^1(\Omega)$ is not a tensor product Hilbert space, in particular $\|\cdot\|_{H^1}$ is not a reasonable crossnorm. Thus, we can not identify $H^1(\Omega)$ with the space of Hilbert Schmidt operators and apply SVD. However, it is known that $H^1(\Omega)$ is isomorph (here written for $d=2$) to the Banach intersection space $$H^1(\Omega_1\times\Omega_2)=H^1(\Omega_1)\otimes L_2(\Omega_2)\cap L_2(\Omega_1)\otimes H^1(\Omega_2)$$ with equivalent norms. Each of the spaces in the intersection is a tensor product Hilbert space where SVD applies. We investigate several approaches to construct low-rank approximations for a function $u\in H^1(\Omega_1\times\Omega_2)$.

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-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 twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals Positive p-summing operators, vector measures and tensor products

1988 ◽  
Vol 31 (2) ◽  
pp. 179-184 ◽  
Author(s):  
Oscar Blasco
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Lattice ◽  
Tensor Products ◽  
Boundary Values ◽  
Vector Measures ◽  
Absolutely Summing Operators

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

Tensor products which do not preserve operator algebras

1990 ◽  
Vol 108 (2) ◽  
pp. 395-403 ◽  
Author(s):  
David P. Blecher
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Operator Algebras ◽  
Tensor Products ◽  
Preserve Operator ◽  
Infinite Dimensional ◽  
Hilbert Space Operators ◽  
Algebra Of Operators

Of late the link between operator algebras and certain tensor products has been reiterated [5]. We prove here that the projective and Haagerup tensor products of two infinite-dimensional C*-algebras is not even topologically isomorphic to an algebra of operators on a Hilbert space. Estimates are given for the distance of the tensor product from such an algebra. Nonetheless with respect to a natural multiplication the Haagerup tensor product of two algebras of Hilbert space operators is completely isometrically isomorphic to an algebra of operators on some B(ℋ).

scholarly journals α-Derivations and their norm in projective tensor products ofΓ-Banach algebras

1998 ◽  
Vol 21 (2) ◽  
pp. 359-368 ◽  
Author(s):  
T. K. Dutta ◽  
H. K. Nath ◽  
R. C. Kalita
Keyword(s):  
Tensor Product ◽  
Arbitrary Element ◽  
Banach Algebras ◽  
Tensor Products ◽  
Projective Tensor Product ◽  
Projective Tensor

Let(V,Γ)and(V′,Γ′)be Gamma-Banach algebras over the fieldsF1andF2isomorphic to a fieldFwhich possesses a real valued valuation, and(V,Γ)⊗p(V′,Γ′), their projective tensor product. It is shown that ifD1andD2areα- derivation andα′- derivation on(V,Γ)and(V′,Γ′)respectively andu=∑1x1⊗y1, is an arbitrary element of(V,Γ)⊗p(V′,Γ′), then there exists anα⊗α′- derivationDon(V,Γ)⊗p(V′,Γ′)satisfying the relationD(u)=∑1[(D1x1)⊗y1+x1⊗(D2y1)]and possessing many enlightening properties. The converse is also true under a certain restriction. Furthermore, the validity of the results‖D‖=‖D1‖+‖D2‖andsp(D)=sp(D1)+sp(D2)are fruitfully investigated.

scholarly journals The “maximal” tensor product of operator spaces

1999 ◽  
Vol 42 (2) ◽  
pp. 267-284 ◽  
Author(s):  
Timur Oikhberg ◽  
Gilles Pisier
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Hilbert Spaces ◽  
Tensor Products ◽  
Operator Space ◽  
Operator Spaces ◽  
Two Factors

In analogy with the maximal tensor product of C*-algebras, we define the “maximal” tensor product E1⊗μE2 of two operator spaces E1 and E2 and we show that it can be identified completely isometrically with the sum of the two Haagerup tensor products: E1⊗hE2 + E2⊗hE1. We also study the extension to more than two factors. Let E be an n-dimensional operator space. As an application, we show that the equality E*⊗μE = E*⊗min E holds isometrically iff E = Rn or E = Cn (the row or column n-dimensional Hilbert spaces). Moreover, we show that if an operator space E is such that, for any operator space F, we have F ⊗min E = F⊗μ E isomorphically, then E is completely isomorphic to either a row or a column Hilbert space.

scholarly journals DAUGAVET PROPERTY IN TENSOR PRODUCT SPACES

2019 ◽  
pp. 1-20 ◽  
Author(s):  
Abraham Rueda Zoca ◽  
Pedro Tradacete ◽  
Ignacio Villanueva
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Symmetric Tensor ◽  
Continuous Functions ◽  
Product Spaces ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions ◽  
Daugavet Property

We 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.

scholarly journals QUANTUM GROUP-TWISTED TENSOR PRODUCTS OF C*-ALGEBRAS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450019 ◽  
Author(s):  
RALF MEYER ◽  
SUTANU ROY ◽  
STANISŁAW LECH WORONOWICZ
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Quantum Group ◽  
Quantum Groups ◽  
Tensor Products ◽  
Group Representations ◽  
Basic Properties ◽  
C Algebras ◽  
Twisted Tensor Product

We put two C*-algebras together in a noncommutative tensor product using quantum group coactions on them and a bicharacter relating the two quantum groups that act. We describe this twisted tensor product in two equivalent ways, based on certain pairs of quantum group representations and based on covariant Hilbert space representations, respectively. We establish basic properties of the twisted tensor product and study some examples.

scholarly journals ON WEYL AND BROWDER SPECTRA OF TENSOR PRODUCTS

2008 ◽  
Vol 50 (2) ◽  
pp. 289-302 ◽  
Author(s):  
C. S. KUBRUSLY ◽  
B. P. DUGGAL
Keyword(s):  
Hilbert Space ◽  
Tensor Product ◽  
Tensor Products ◽  
Hilbert Space Operators ◽  
Weyl Spectrum

AbstractLet A and B be Hilbert space operators. In this paper we explore the structure of parts of the spectrum of the tensor product A ⊗ B, and conclude some properties that follow from such a structure. We give conditions on A and B ensuring that σw(A ⊗ B) =σw(A)ċσ(B) ∪ σ(A)ċσw(B), where σ(ċ) and σw(ċ) stand for the spectrum and Weyl spectrum, respectively. We also investigate the problem of transferring Weyl and Browder's theorems from A and B to their tensor product A⊗B.

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