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.     

Monotone norms

Summary The idea of norms monotone in the nonnegative orthant is extended to partially ordered finite dimensional real spaces. Relations between the monotonicity of a norm and of the dual and the subordinate matrix norm are investigated.     

Estimates for norms of resolvents of~operators on tensor products of Hilbert spaces

A class of linear operators on tensor products of Hilbert spaces is considered. That class contains integro-differential operators arising in various applications. Estimates for the norm of the resolvent of considered operators are derived. By virtue of the obtained estimates, the spectrum of perturbed operators is investigated. These results are new even in the finite-dimensional case. Applications to integro-differential operators are also discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Contractive homomorphisms and tensor product norms

For any complex domain , one can ask if all contractive algebra homomorphisms ofA() (into the algebra of Hilbert space operators) are completely contractive or not. By Ando's Theorem, this has an affirmative answer for =D ², the bi-disc-while the answer is unknown for =D ², the unit ball of ² with ¹-norm. In this paper, we consider a special class of homomorphisms associated with any bounded complex domain; this well known construct generalizes Parrott's example. Our question has an affirmative answer for homomorphisms in this class with = (¹(2))₁. We show that there are many domains in ² for which the question can be answered in the affirmative by reducing it to that of =D ² or (¹(2))₁. More generally, the question for an arbitrary can often be reduced to the case of the unit ball of an associated finite dimensional Banach space. If we restrict attention to a smaller subclass of homomorphisms the question for a Banach ball becomes equivalent to asking whether in the analogue of Grothendieck's inequality, in this Banach space, restricted to positive operators, the best constant is = 1 or not. We show that this is indeed the case for =D ²,D ³ or the dual balls, but not forD ⁿ or its dual forn4. Thus we isolate a large class of homomorphisms ofA(D ³) for which contractive implies completely contractive. This has many amusing relations with injective and projective tensor product norms and with Parrott's example.     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

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.     

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.     

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.