Grothendieck Inequality Research Articles

Vector integration and the Grothendieck inequality

We relate the Grothendieck inequality to the theory of vector measures and show that the integral of an inner product with respect to a bimeasure can be computed in an iterative way. We then show an application to the theory of bounded linear operators.

Representation of certain homogeneous Hilbertian operator spaces and applications

Following Grothendieck's characterization of Hilbert spaces we consider operator spaces $F$ such that both $F$ and $F^*$ completely embed into the dual of a C*-algebra. Due to Haagerup/Musat's improved version of Pisier/Shlyakhtenko's Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum $C\oplus R$ of the column and row spaces (the corresponding class being denoted by $QS(C\oplus R)$). We first prove a representation theorem for homogeneous $F\in QS(C\oplus R)$ starting from the fundamental sequences defined by column and row norms of unit vectors. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of $F$ and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an $n$-dimensional subspace $F_n$ of $F$ involving the fundamental sequences. Similarly, we have formulas for the the projection (=injectivity) constant of $F_n$. They also permit us to determine the completely 1-summing maps in Effros and Ruan's sense between two homogeneous spaces $E$ and $F$ in $QS(C\oplus R)$.

SDP gaps and UGC-hardness for Max-Cut-Gain

$ \newcommand{\eps}{\epsilon} $ Given a graph with maximum cut of (fractional) size $c$, the semidefinite programming (SDP)-based algorithm of Goemans and Williamson is guaranteed to find a cut of size at least $.878 \cdot c$. However this guarantee becomes trivial when $c$ is near $1/2$, since making random cuts guarantees a cut of size $1/2$ (i.e., half of all edges). A few years ago, Charikar and Wirth (analyzing an algorithm of Feige and Langberg) showed that given a graph with maximum cut $1/2 + \eps$, one can find a cut of size $1/2 + \Omega(\eps/\log(1/\eps))$. The main contribution of our paper is twofold: We give a natural and explicit $1/2 + \eps$ vs. $1/2 + O(\eps/\log(1/\eps))$ integrality gap for the Max-Cut SDP based on Euclidean space with the Gaussian probability distribution. This shows that the SDP-rounding algorithm of Charikar-Wirth is essentially best possible. We show how this SDP gap can be translated into a Long Code test with the same parameters. This implies that beating the Charikar-Wirth guarantee with any efficient algorithm is NP-hard, assuming the Unique Games Conjecture (UGC). This result essentially settles the asymptotic approximability of Max-Cut, assuming UGC. Building on the first contribution, we show how “randomness reduction” on related SDP gaps for the Quadratic-Programming problem lets us make the $\Omega(\log(1/\eps))$ gap as large as $\Omega(\log n)$ for $n$-vertex graphs. In addition to optimally answering an open question of Alon, Makarychev, Makarychev, and Naor, this technique may prove useful for other SDP gap problems. Finally, illustrating the generality of our second contribution, we also show how to translate the Davie—Reeds SDP gap for the Grothendieck Inequality into a UGC-hardness result for computing the $\| \cdot \|_{\infty \mapsto 1}$ norm of a matrix.

A new multilinear insight on Littlewood's 4/3-inequality

We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in ℓ p -spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous ℓ p -valued polynomials on c 0 .

Grothendieck type inequalities

We present here the analogue of Grothendieck inequality for positive linear forms. We obtain upper and lower bounds of L p 0 < p ≤ ∞ , type, which all lead to sharp inequalities.

A Maurey type result for operator spaces

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C ( K ) and ℓ 2 is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2) (2005) 225–286] that the operator space analogue fails. Not every cb-map v : K → OH is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map v : K → OH is ( q , cb ) -summing for any q > 2 and hence admits a factorization ‖ v ( x ) ‖ ⩽ c ( q ) ‖ v ‖ cb ‖ a x b ‖ q with a , b in the unit ball of the Schatten class S 2 q .

New Bell inequalities for the singlet state: Going beyond the Grothendieck bound

Contemporary versions of Bell’s argument [Physics (Long Island City, N.Y.) 1, 195 (1964)] against local hidden variable (LHV) theories are based on the Clauser-Horne-Shimony-Holt (CHSH) [Phys. Rev. Lett. 23, 880 (1969)] inequality and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper, we derive two-particle Bell inequalities for traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constants both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations and applying a recent result of Alon et al. [Invent. Math. 163, 499 (2006)], we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to infinity as the dimension and number of measurements grow. Technically, these result are possible because perfect correlations (or anticorrelations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%.

The Grothendieck constant of random and pseudo-random graphs

The Grothendieck constant of a graph G = ( V , E ) is the least constant K such that for every matrix A : V × V → R , max f : V → S | V | − 1 ∑ { u , v } ∈ E A ( u , v ) ⋅ 〈 f ( u ) , f ( v ) 〉 ≤ K max ϵ : V → { − 1 , + 1 } ∑ { u , v } ∈ E A ( u , v ) ⋅ ϵ ( u ) ϵ ( v ) . The investigation of this parameter, introduced in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486–493 (Also: Invent. Math. 163 (2006) 499–522)], is motivated by the algorithmic problem of maximizing the quadratic form ∑ { u , v } ∈ E A ( u , v ) ϵ ( u ) ϵ ( v ) over all ϵ : V → { − 1 , 1 } , which arises in the study of correlation clustering and in the investigation of the spin glass model. In the present note we show that for the random graph G ( n , p ) the value of this parameter is, almost surely, Θ ( log ( n p ) ) . This settles a problem raised in [N. Alon, K. Makarychev, Y. Makarychev, A. Naor, Quadratic forms on graphs, in: Proc. of the 37th ACM STOC, ACM Press, Baltimore, 2005, pp. 486–493 (Also: Invent. Math. 163 (2006) 499–522)]. We also obtain a similar estimate for regular graphs in which the absolute value of each nontrivial eigenvalue is small.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Quantum correlation and Grothendieck's constant

The best upper bound for the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of $\pm 1$ valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck's constant ($\mathcal{K}_{G}\approx 1.73\pm0.06$) as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four $\pm 1$ valued observables on both sites of a correlation experiment and show that the quantum violation approaching 1.58. Moreover, we estimate the maximal quantum violation of a correlation experiment for large and equal number of $\pm 1$ valued observables on both sites. In this case, the maximal quantum violation converges to $\sqrt{3}\approx1.73$ for very large $n$, which coincides with the approximate value of Grothendieck's constant.

Grothendieck’s constant and local models for noisy entangled quantum states

We relate the nonlocal properties of noisy entangled states to Grothendieck's constant, a mathematical constant appearing in Banach space theory. For two-qubit Werner states $\rho^W_p=p \proj{\psi^-}+(1-p){\one}/{4}$, we show that there is a local model for projective measurements if and only if $p \le 1/K_G(3)$, where $K_G(3)$ is Grothendieck's constant of order 3. Known bounds on $K_G(3)$ prove the existence of this model at least for $p \lesssim 0.66$, quite close to the current region of Bell violation, $p \sim 0.71$. We generalize this result to arbitrary quantum states.

Operator-space Grothendieck inequalities for noncommutative Lp-spaces

We prove the operator-space Grothendieck inequality for bilinear forms on subspaces of noncommutative Lp-spaces with 2<p<∞. One of our results states that given a map u:E→F*, where E,F⊂Lp(M) (2<p<∞, M being a von Neumann algebra), u is completely bounded if and only if u factors through a direct sum of a p-column space and a p-row space. We also obtain several operator-space versions of the classical little Grothendieck inequality for maps defined on a subspace of a noncommutative Lp-space (2<p<∞) with values in a q-column space for every q∈[p',p] (p' being the index conjugate to p). These results are the Lp-space analogues of the recent works on the operator-space Grothendieck theorems by Pisier and Shlyakhtenko. The key ingredient of our arguments is some Khintchine-type inequalities for Shlyakhtenko's generalized circular systems. One of our main tools is a Haagerup-type tensor norm that turns out to be particularly fruitful when applied to subspaces of noncommutative Lp-spaces (2<p<∞). In particular, we show that the norm dual to this tensor norm, when restricted to subspaces of noncommutative Lp-spaces, is equal to the factorization norm through a p-row space

Approximating the Cut-Norm via Grothendieck's Inequality

The cut-norm $||A||_C$ of a real matrix $A=(a_{ij})_{i\in R,j\in S}$ is the maximum, over all $I \subset R$, $J \subset S$, of the quantity $|\sum_{i \in I, j\in J} a_{ij}|$. This concept plays a major role in the design of efficient approximation algorithms for dense graph and matrix problems. Here we show that the problem of approximating the cut-norm of a given real matrix is MAX SNP hard, and we provide an efficient approximation algorithm. This algorithm finds, for a given matrix $A=(a_{ij})_{i\in R,j\in S}$, two subsets $I \subset R$ and $J \subset S$, such that $|\sum_{i \in I, j\in J} a_{ij}| \geq \rho ||A||_C$, where $\rho>0$ is an absolute constant satisfying $\rho >0.56$. The algorithm combines semidefinite programming with a rounding technique based on Grothendieck's inequality. We present three known proofs of Grothendieck's inequality, with the necessary modifications which emphasize their algorithmic aspects. These proofs contain rounding techniques which go beyond the random hyperplane rounding of Goemans and Williamson [J. ACM, 42 (1995), pp. 1115-1145], allowing us to transfer various algorithms for dense graph and matrix problems to the sparse case.

Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in L loc 2( X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ť θ = T if the corresponding Hilbert space is contained in L loc 2( X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of L loc 2 ( X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.

Quadratic forms on graphs

We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→ℝ, $$\sup_{f:V\to{S}^{|V|-1}}\sum_{\{u,v\}\in{E}} A(u,v)\cdot\langle{f(u),f(v)}\rangle\le{K}\sup_{\varphi:V\to\{-1,+1\}}\sum_{\{u,v\}\in{E}}A(u,v)\cdot\varphi(u)\varphi(v).$$ The classical Grothendieck inequality corresponds to the case of bipartite graphs, but the case of general graphs is shown to have various algorithmic applications. Indeed, our work is motivated by the algorithmic problem of maximizing the quadratic form ∑{u,v}∈E A(u,v)ϕ(u)ϕ(v) over all ϕ:V→{-1,1}, which arises in the study of correlation clustering and in the investigation of the spin glass model. We give upper and lower estimates for the integrality gap of this program. We show that the integrality gap is $O(\log\vartheta(\overline{G}))$ , where $\vartheta(\overline{G})$ is the Lovasz Theta Function of the complement of G, which is always smaller than the chromatic number of G. This yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of graphs G. We also show that the maximum possible integrality gap is always at least Ω(log ω(G)), where ω(G) is the clique number of G. In particular it follows that the maximum possible integrality gap for the complete graph on n vertices with no loops is Θ(logn). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is Θ(logn). The lower bound for the complete graph improves a result of Kashin and Szarek on Gram matrices of uniformly bounded functions, and settles a problem of Megretski and of Charikar and Wirth.

Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’

We use Voiculescu's concept of free probability to construct a completely isomorphic embedding of the operator space OH in the predual of a von Neumann algebra. We analyze the properties of this embedding and determine the operator space projection constant of OH_n: It is of the order (n/1+ln n)^{1/2} The lower estimate is a recent result of Pisier and Shlyakhtenko that improves an estimate of order 1/(1+ln n) of the author. The additional factor 1/(1+ln n)^{1/2} indicates that the operator space OH_n behaves differently than its classical counterpart l_2^n. We give an application of this formula to positive sesquilinear forms on B(H). This leads to logarithmic characterization of C*-algebras with the weak expectation property introduced by Lance.

New advances on the Grothendieck's inequality problem for bilinear forms on JB*-triples

The results known as Grothendieck’s inequalities began with the famous paper [8] in which A. Grothendieck proved the so-called “Grothendieck’s inequalities” for commutative C*-algebras. These inequalities were generalized by G. Pisier [18] and U. Haagerup [10, 9] to the setting of C*-algebras. Every C*-algebra belongs to a more general class of Banach spaces known as JB*-triples (see definition and examples below). JB*-triples were introduced by Kaup [14] in the study of bounded symmetric domains in complex Banach spaces. The class of JB*-triples has been intensively developed in the last twenty years. In the setting of JB*-triples, Grothendieck’s inequalities were studied by T. Barton and Y. Friedman [1], C.-H. Chu, B. Iochum and G. Loupias [3], A. M. Peralta [15] and A. M. Peralta and A. Rodŕiguez Palacios [16, 17]. The natural prehilbertian seminorms associated derived from states in a C*-algebra do not make sense in a JB*-triple because the latter needs not have, in general, a natural order structure. In the setting of JB*-triples, the prehilbertian seminorms associated to norm-one functionals are constructed as follows: Let φ be a norm-one element in the dual space of a JB*-triple

Operator space structure and amenability for Figà-Talamanca–Herz algebras

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p, p′∈(1,∞) with 1 p + 1 p′ =1 , we use the operator space structure on CB( COL(L p′(G))) to equip the Figà-Talamanca–Herz algebra A p ( G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p⩽ q⩽2 or 2⩽ q⩽ p and amenable G, the canonical inclusion A q ( G)⊂ A p ( G) is completely bounded (with cb-norm at most K G 2 , where K G is Grothendieck's constant). As an application, we show that G is amenable if and only if A p ( G) is operator amenable for all—and equivalently for one— p∈(1,∞); this extends a theorem by Ruan.

Grothendieck's Inequalities for Real and Complex JBW*-Triples

We prove that, if M > 4 ( 12 3 ) and ɛ > 0, if V and W are complex JBW*-triples (with preduals V* and W*, respectively), and if U is a separately weak*-continuous bilinear form on V × W, then there exist norm-one functionals ϕ1, ϕ2 ∈ V* and ψ1, ψ2 ∈ W* satisfying | U ( x , y ) | ⩽ M ∥ U ∥ ( ∥ x ∥ φ 2 2 ε 2 ∥ x ∥ φ 1 2 ) 1 2 ( ∥ y ∥ ψ 2 2 ε 2 ∥ y ∥ ψ 1 2 ) 1 2 for all (x, y) ∈ V × W. Here, for a norm-one functional ϕ on a complex JB*-triple V, |·|ϕ stands for the prehilbertian seminorm on V associated to ϕ given by ∥ x ∥ φ 2 : = φ { x , x , z } for all x ∈ W, where z ∈ V** satisfies ϕ z = |z| = 1. We arrive at this form of ‘Grothendieck's inequality’ through results of C.-H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*-triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*-triples. 2000 Mathematical Subject Classification: 17C65, 46K70, 46L05, 46L10, 46L70.

A vector-valued Grothendieck inequality with an application to (p, q)-completely bounded operators

A vector-valued Grothendieck inequality with an application to (p, q)-completely bounded operators