Tensor Products Of Functions Research Articles

Partial tensor-product functors and crossed-product functors

For a given discrete group G G , we apply results of Kirchberg on exact and injective tensor products of C ∗ C^* -algebras to give an explicit description of the minimal exact correspondence crossed-product functor and the maximal injective crossed-product functor for G G in the sense of Buss, Echterhoff and Willett. In particular, we show that the former functor dominates the latter.

On singular equivalences of Morita type with level and Gorenstein algebras

Rickard proved that for certain self-injective algebras, a stable equivalence induced from an exact functor is a stable equivalence of Morita type, in the sense of Broué. In this paper we study singular equivalences of finite-dimensional algebras induced from tensor product functors. We prove that for certain Gorenstein algebras, a singular equivalence induced from tensoring with a suitable complex of bimodules induces a singular equivalence of Morita type with level, in the sense of Wang. This recovers Rickard's theorem in the self-injective case.

Frobenius reciprocity and the Haagerup tensor product

In the context of operator-space modules over C ∗ C^* -algebras, we give a complete characterisation of those C ∗ C^* -correspondences whose associated Haagerup tensor product functors admit left adjoints. The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors. As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H H of a locally compact group G G admits a left adjoint in this setting if and only if H H is cocompact in G G . The adjoint functor is given by the Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule.

Generalized polynomial approximation and growth estimates of entire functions of two complex variables in Banach spaces

Infinite dimensional subspaces, which consists of forms involving tensor products of functions of fewer variables, play an important role in the theory of approximation of functions of several real variables. Similar technique can be applied to problems of approximation of analytic functions of several complex variables. In the present paper we study the generalized polynomial approximation of entire functions of two complex variables in Banach spaces. Moreover, limit equations relating growth parameters order and type with above approximation errors in Banach spaces have been investigated.

Braided quantum SU(2) groups

We construct a family of q -deformations of SU(2) for complex parameters q \neq 1 . For real q , the deformation coincides with the compact quantum SU _q (2) group. For q \notin \mathbb R , SU _q (2) is only a braided compact quantum group with respect to a twisted tensor product functor for C*-algebras with an action of the circle group.

Relative left derived functors of tensor product functors

We introduce and study the relative left derived functor Torn(F,F′)(−,−) in the module category, which unifies several related left derived functors. Then we give some criteria for computing the F-resolution dimensions of modules in terms of the properties of TorTorn(F,F′)(−,−). We also construct a complete and hereditary cotorsion pair relative to balanced pairs. Some known results are obtained as corollaries.

Coproduct of 2-crossed modules: applications to a definition of a tensor product for 2-crossed complexes

In this paper it is shown how to construct finite coproducts in the category of 2-crossed modules. We give explicit descriptions of the pullback functor and induced functor along a morphism of precrossed modules which are used to built the coproduct of 2-crossed modules. As an application we give a definition and construction of a possible tensor product functor for 2-crossed complexes.

A CONTINUUM OF C*-NORMS ON (H) ⊗ (H) AND RELATED TENSOR PRODUCTS

AbstractFor any pairM,Nof von Neumann algebras such that the algebraic tensor productM⊗Nadmits more than one C*-norm, the cardinal of the set of C*-norms is at least 2ℵ0. Moreover, there is a family with cardinality 2ℵ0of injective tensor product functors for C*-algebras in Kirchberg's sense. Let${\mathbb B}$=∏nMn. We also show that, for any non-nuclear von Neumann algebraM⊂${\mathbb B}$(ℓ2), the set of C*-norms on${\mathbb B}$⊗Mhas cardinality equal to 22ℵ0.

The pivotal cover and Frobenius–Schur indicators

In this paper, we introduce the notion of the pivotal cover Cpiv of a left rigid monoidal category C to develop a theoretical foundation for the theory of Frobenius–Schur (FS) indicators in “non-pivotal” settings. For an object V∈Cpiv, the (n,r)-th FS indicator νn,r(V) is defined by generalizing that of an object of a pivotal monoidal category. This notion gives a categorical viewpoint to some recent results on generalizations of FS indicators.Based on our framework, we also study the FS indicators of the “adjoint object” in a finite tensor category, which can be considered as a generalization of the adjoint representation of a Hopf algebra. The indicators of this object closely relate to the space of endomorphisms of the iterated tensor product functor.

BALANCE FOR RELATIVE HOMOLOGY WITH RESPECT TO SEMIDUALIZING MODULES

We derive in the paper the tensor product functor -<TEX>${\otimes}_R$</TEX>- by using proper <TEX>$\mathcal{GP}_C$</TEX>-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of <TEX>$\mathcal{G}_C$</TEX>-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.

APPENDIX: EXACTNESS OF THE MAXIMAL TENSOR PRODUCT OF REAL C*-ALGEBRAS

This appendix to article [M. Karoubi and M. Wodzicki, Algebraic and Hermitian K-Theory of stable Algebras (in this volume)] provides a self-contained proof of the exactness of the maximal tensor product functor for real C*-algebras.

Preservation of quasi-isomorphisms of complexes

We consider the preservation property of the homomorphism and tensor product functors for quasi-isomorphisms and equivalences of complexes. Let X and Y be two classes of R-modules with Ext〉-1(X, Y ) = 0 for each object X ∈ X and each object Y ∈ Y. We show that if A,B ∈ C ⊐(R) are X-complexes and U, V ∈ C ⊏(R) are Y-complexes, then Open image in new window ; Open image in new window . As an application, we give a sufficient condition for the Hom evaluation morphism being invertible.

Hom functors and tensor product functors in fuzzy S-act category

In this paper, we construct the fuzzy S-act HomS(μA, νB), where μA and νB are two fuzzy S-acts. We also prove that HomS(ξp, −) preserves short exact sequence if and only if \(\xi_P\cong \coprod\nolimits_{i\in i}0_{Se_i},\) where ei ∈ E(S). Moreover, we give the definition of tensor product of two fuzzy S-acts and get some properties of tensor product.

Mars approach for global sensitivity analysis of differential equation models with applications to dynamics of influenza infection.

Differential equation models are widely used for the study of natural phenomena in many fields. The study usually involves unknown factors such as initial conditions and/or parameters. It is important to investigate the impact of unknown factors (parameters and initial conditions) on model outputs in order to better understand the system the model represents. Apportioning the uncertainty (variation) of output variables of a model according to the input factors is referred to as sensitivity analysis. In this paper, we focus on the global sensitivity analysis of ordinary differential equation (ODE) models over a time period using the multivariate adaptive regression spline (MARS) as a meta model based on the concept of the variance of conditional expectation (VCE). We suggest to evaluate the VCE analytically using the MARS model structure of univariate tensor-product functions which is more computationally efficient. Our simulation studies show that the MARS model approach performs very well and helps to significantly reduce the computational cost. We present an application example of sensitivity analysis of ODE models for influenza infection to further illustrate the usefulness of the proposed method.

Proper general decomposition (PGD) for the resolution of Navier–Stokes equations

In this work, the PGD method will be considered for solving some problems of fluid mechanics by looking for the solution as a sum of tensor product functions. In the first stage, the equations of Stokes and Burgers will be solved. Then, we will solve the Navier–Stokes problem in the case of the lid-driven cavity for different Reynolds numbers (Re=100, 1000 and 10,000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.

Proper Generalized Decomposition method for incompressible flows in stream-vorticity formulation

In this work, the Proper Generalized Decomposition (PGD) method will be considered in order to solve Navier-stokes equations with a stream-vorticity formulation by looking for the solution as a sum of tensor product functions. In the first stage, PGD will be applied to a model equation in order to test the capacity of the method to treat some timedependent problem. Then, we will solve the Navier-Stokes problem in the case of the liddriven cavity for different Reynolds numbers (Re = 100, 1000 and 10000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.

Proper Generalized Decomposition method for incompressible flows in stream-vorticity formulation

In this work, the Proper Generalized Decomposition (PGD) method will be considered in order to solve Navier-stokes equations with a stream-vorticity formulation by looking for the solution as a sum of tensor product functions. In the first stage, PGD will be applied to a model equation in order to test the capacity of the method to treat some timedependent problem. Then, we will solve the Navier-Stokes problem in the case of the liddriven cavity for different Reynolds numbers (Re = 100, 1000 and 10000). Finally, the PGD method will be compared to the standard resolution technique, both in terms of CPU time and accuracy.

One more pathology of [formula omitted]-algebraic tensor products

We define a collection of tensor product norms for C ∗ -algebras and show that a symmetric tensor product functor on the category of separable C ∗ -algebras need not be associative.

Subpullback flat S-posets need not be subequalizer flat

If S is a monoid, a right S-act AS is a set A, equipped with a “right S-action” A×S→A sending the pair (a,s)∈ A×S to as, that satisfies the conditions (i) a(st)=(as)t and (ii) a1=a for all a∈A and s,t∈S. If, in addition, S is equipped with a compatible partial order and A is a poset, such that the action is monotone (when A×S is equipped with the product order), then AS is called a right S-poset. Left S-acts and S-posets are defined analogously. For a given S-act (resp. S-poset) a tensor product functor AS⊗− from left S-acts to sets (resp. left S-posets to posets) exists, and AS is called pullback flat or equalizer flat (resp. subpullback flat or subequalizer flat) if this functor preserves pullbacks or equalizers (resp. subpullbacks or subequalizers). By analogy with the Lazard-Govorov Theorem for R-modules, B. Stenstrom proved in 1971 that an S-act is isomorphic to a directed colimit of finitely generated free S -acts if and only if it is both pullback flat and equalizer flat. Some 20 years later, the present author showed that, in fact, pullback flatness by itself is sufficient. (A new, more direct proof of that result is contained in the present article.) In 2005, Valdis Laan and the present author obtained a version of the Lazard-Govorov Theorem for S-posets, in which subpullbacks and subequalizers now assume the role previously played by pullbacks and equalizers. The question of whether subpullback flatness implies subequalizer flatness remained unsolved. The present paper provides a negative answer to this question.

A generalization of a theorem of A. Grothendieck

In this article we characterize the quasi-barrelledness of the projective tensor product of a coechelon space of type one k 1(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k 1(A) ·: ℱ → ℒ︁, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor1π (k 1(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)