Grothendieck Inequality Research Articles

Local properties of accessible injective operator ideals

In addition to Pisier's counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a “good behaviour” of trace duality, which is canon-ically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible injective Banach ideals, where we also recognize the appearance of the ideal of absolutely summing operators (prop. 4.1). By the famous Grothendieck inequality, every operator from L 1 to a Hilbert space is absolutely summing, and therefore our search for such ideals will be directed towards Hilbert space factorization—via an operator version of Grothendieck's inequality (lemma 4.2). As a consequence, we obtain a class of injective ideals, which are “quasi-accessible”, and with the help of tensor stability, we improve the corresponding norm inequalities, to get accessibility (theorem 4.1 and 4.2). In the last chapter of this paper we give applications, which are implied by a non-trivial link of the above mentioned considerations to normed products of operator ideals.

Contractive homomorphisms and tensor product norms

For any complex domain Ω, one can ask if all contractive algebra homomorphisms ofA(Ω) (into the algebra of Hilbert space operators) are completely contractive or not. By Ando's Theorem, this has an affirmative answer for Ω =D 2, the bi-disc-while the answer is unknown for Ω =D 2, the unit ball of ℂ2 with ℓ1-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 Ω = (ℓ1(2))1. We show that there are many domains in ℓ2 for which the question can be answered in the affirmative by reducing it to that of Ω =D 2 or (ℓ1(2))1. More generally, the question for an arbitrary Ω 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 Ω =D 2,D 3 or the dual balls, but not forD n or its dual forn≥4. 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.

A specific form of Grothendieck's inequality for the two-dimensional case, with applications to C*-algebras

We characterize bilinear forms V on such that V(e, e) = ‖V‖ = 1 in terms of their matrices. For such V we prove that |V(x, y)|2≦φ(|x|2)ψ(|y|2) for all x, y, where φ(x)= V(x, e), ψ(y) = V(e, y). Some other properties of such forms are given.

Harmonizability,V-boundedness, (2,p)-boundedness of stochastic processes

Some new classes of discrete time non-stationary processes, related to the harmonizable andV-bounded classes, are introduced. A few characterizations are obtained which, in turn, unify theV-bounded theory. Our main results depend on a special form of Grothendieck's inequality.

Multi-linear measure theory and multiple stochastic integration

Stochastic integration in a Lebesgue Stieltjes sense is developed in a general context of multi-linear measure theory based on the Fundamental Grothendieck Inequality and Factorization Theorem.

Completely Bounded Multilinear Maps and Grothendieck's Inequality

Bulletin of the London Mathematical SocietyVolume 20, Issue 6 p. 606-612 Notes and papers Completely Bounded Multilinear Maps and Grothendieck's Inequality R. R. Smith, R. R. Smith Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USASearch for more papers by this author R. R. Smith, R. R. Smith Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USASearch for more papers by this author First published: November 1988 https://doi.org/10.1112/blms/20.6.606Citations: 5AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volume20, Issue6November 1988Pages 606-612 RelatedInformation

A new upper bound for the complex Grothendieck constant

Let ϕ denote the real function $$\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1$$ and letK G C be the complex Grothendieck constant. It is proved thatK G C ≦8/π(k 0+1), wherek 0 is the (unique) solution to the equationϕ(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k 0+1) ≈ 1.40491. The previously known upper bound isK G C ≦e 1−y ≈ 1.52621 obtained by Pisier in 1976.

An Elementary Proof of the Grothendieck Inequality

An elementary proof of the Grothendieck inequality is given.

Random Clarkson inequalities and LP versions of Grothendieck's inequality

AbstractIn a recent paper KATO [3] used the LITTLEWOOD matrices to generalise CLARKSON'S inequalities. Our first aim is to indicate how KATO'S result can be deduced from a neglected version of the HAUSDORFF‐YOUNG inequality which was proved by WELLS and WILLIAMS [12]. We next establish „random CLARKSON inequalities”︁. These show that the expected behaviour of matrices whose coefficients are random ± 1′s is, as one might expect, the same as the behaviour that KATO observed in the LITTLEWOOD matrices. Finally we show how sharp Lp versions of GROTHENDIECK'S inequality can be obtained by combining a KATO‐like result with a theorem of BENNETT [1] on SCHUR multipliers.

An elementary proof of the Grothendieck inequality

An elementary proof of the Grothendieck inequality is given.

Bimeasure algebras on locally compact groups

For locally compact groups G and H, let BM( G, H) denote the Banach space of bounded bilinear forms on C 0( G) × C 0( H). Using a consequence of the fundamental inequality of A. Grothendieck. a multiplication and an adjoint operation are introduced on BM( G, H) which generalize the convolution structure of M( G × H) and which make BM( G, H) into a K G 2-Banach ∗ -algebra, where K G is Grothendieck's universal constant. Various topics relating to the ideal structure of BM( G, H) and the lifting of unitary representations of G × H to ∗ -representations of BM( G, H) are investigated.

The interpolation proof of Grothendieck's inequality

This note is an exposition of the simple and elegant approach to Grothendieck's inequality given in [2] and lsqb;4], with one further simplification. The process of factorizing through L2 ([2], p. 21) introduces a factor of into the final constant. We show that this step can be avoided.

The Grothendieck inequality for bilinear forms on C∗-algebras

The following generalization of Grothendieck's inequality is proved: For any bounded bilinear form V on a pair of C∗-algebras A, B, there exist two states ϕ 1, ϕ 2 on A and two states ψ 1, ψ 2 on B, such that |V(x,y)|⩽‖V‖(ϕ 1(x ∗x)+ϕ 2(xx 2)) 1 2 (φ 1(y ∗y)+ϕ 2(yy 2)) 1 2 for all x ϵ A and all y ϵ B. An inequality of this type was proved a few years ago by Pisier in the case where one of the C∗-algebras has the bounded approximation property. It follows from the above inequality that any bounded linear map T of a C∗-algebra into the dual of a C∗-algebra has a factorization T = R ∘ S through a Hilbert space, such that ‖ R ‖ ‖ S ‖ ⩽ 2 ‖ T ‖.

Bimeasure algebras on LCA groups

For locally compact abelian groups Gλ and G2, with character groups Γ, and Γ2, respectively, let BM(GU G2) denote the Banach space of bounded bilinear forms on CO(GX) X C0(G2). Using a consequence of the fundamental inequality of A. Grothendieck, a multiplication and an adjoint operation are introduced on BM{Gλ9 G2) which generalize the convolution structure of M(G X H) and which make BM(GU G2) into a Λ^-Banach *-algebra, where KG is Grothendieck's universal constant. The Fourier transforms of elements of BM(GUG2) are defined and characterized in terms of certain unitary representations of Γ, and Γ2. Various aspects of the harmonic analysis of the algebras BM(GX, G2) are studied.

Polarization and the two-dimensional Grothendieck inequality

Throughout this paper all scalars and vector spaces will be assumed to be complex unless there are specific indications to the contrary.

Banach Lattices and Extensions of Grothendieck's Inequality

Banach Lattices and Extensions of Grothendieck's Inequality

Interpolation sets and extensions of the Grothendieck inequality

Interpolation sets and extensions of the Grothendieck inequality

Multidimensional extensions of the Grothendieck inequality and applications

Multidimensional extensions of the Grothendieck inequality and applications

Grothendieck's theorem for noncommutative C∗-algebras, with an Appendix on Grothendieck's constants

We study a conjecture of Grothendieck on bilinear forms on a C ∗-algebra Ol . We prove that every “approximable” operator from Ol into Ol ∗ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendieck's theorem. These results enable us to prove a conjecture of Ringrose on operators on a C ∗-algebra. In the Appendix, we present a new proof of Grothendieck's inequality which gives an improved upper bound for the so-called Grothendieck constant k G .

A proof of the Grothendieck inequality

The fundamental constant of Grothendieck's inequality, defined below, was shown by Grothendieck to be less than sinh π/2=2.301+. We improve the bound slightly, and show that for the positive definite case π/2 suffices.