Graded tensor products

We study the polynomial identities of regular algebras, introduced in [A. Regev, T. Seeman, Z₂-graded tensor products of P.I. algebras, J. Algebra 291 (2005) 274-296]. For example, a finite-dimensional algebra is regular if it has a basis whose multiplication table satisfies some commutation relations. The matrix algebra M_n(F) over the field F is regular, which is closely related to M_n(F) being Z_n-graded. We study the polynomial identities of various types of tensor products of such algebras. In particular, using the theory of Hopf algebras, we prove a far reaching extension of the A⊗B theorem for Z₂-graded PI algebras.     

Simple tensor products

Let ℱ be the category of finite-dimensional representations of an arbitrary quantum affine algebra. We prove that a tensor product S ₁ ⊗⋅⋅⋅⊗ S _N of simple objects of ℱ is simple if and only S _i ⊗ S _j is simple for any i<j.     

Unconditionality in tensor products

It is proved that in order to study unconditional structures in tensor products of finite dimensional Banach spaces it is enough to consider a certain basis. This result is applied to spaces ofp-absolutely summing operators showing their “bad” structure.     

Groupoid normalizers of tensor products

We consider an inclusion B⊆M of finite von Neumann algebras satisfying B^′∩M⊆B. A partial isometry v∈M is called a groupoid normalizer if vBv^∗,v^∗Bv⊆B. Given two such inclusions Bi⊆Mi, i=1,2, we find approximations to the groupoid normalizers of in , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis , i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v∈M satisfying vBv^∗⊆B and v^∗v,vv^∗∈B.     

Algebraic identity for the Schouten tensor and bi-Hamiltonian systems

A (0,3)-tensor Tijk is introduced in an invariant form. Algebraic identities are derived that connect the Schouten (2,1)-tensor and tensor Tijk with the Nijenhuis tensor . Applications to the bi-Hamiltonian dynamical systems are presented.     

Grothendieck operators on tensor products

We prove that for Banach spaces and operators , the tensor product is a Grothendieck operator, provided is a Grothendieck operator and is compact.

     

Monotone norms and tensor products

Various refinements of orthant monotonicity for norms are studied. A partial order is induced in the tensor product of two partially ordered vector spaces. The induced norm in the tensor product is shown to be orthant monotone in certain cases.     

Lattice tensor products. IV

In 1999, for lattices A and B, G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B ^A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A?B.     

Lattice tensor products. II

G. Grätzer and F. Wehrung has recently introduced the lattice tensor product, A?B, of the lattices A and B. In this note, for a finite lattice A and an arbitrary lattice B, we compute the ideal lattice of A?B, obtaining the isomorphism Id(A?B)≌A?Id B. This generalizes an earlier result of G. Grätzer and F. Wehrung proving this isomorphism for A = M_3 and B n-modular. We prove this isomorphism by utilizing the coordinatization of A?B introduced in Part I of this paper.     

Projective modules and tensor products

In this paper we analyze the problem of transforming one path in Rⁿ to another by means of three geometric operations. The problem is approached in two different ways: via the theory of δ-indecomposable semigroups, and by means of combinatorics.     

Monotone norms and tensor products

Various refinements of orthant monotonicity for norms are studied. A partial order is induced in the tensor product of two partially ordered vector spaces. The induced norm in the tensor product is shown to be orthant monotone in certain cases.     

Ergodic theorems for tensor products

An ergodic theorem is proved for tensor products of Banach spaces. As a special case, an ergodic theorem is proved for vector-valued L^p-spaces. This theorem generalizes results of Aribaud, J. Funct. Anal.5 (1970), 395–411, and Dinculeanu, J. Funct. Anal.12 (1973), 229–235.