Grothendieck Type Research Articles

Locally Coherent Exact Categories

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Št’ovíček. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

Strongly Mackey topologies and Radon vector measures

Let X be a topological Hausdorff space and \mathcal B o be the \sigma -algebra of Borel sets in X . Let B(\mathcal B o) be the space of all bounded \mathcal B o -measurable scalar functions on X , equipped with the Mackey topology \tau(B(\mathcal B o),M(X)) , where M(X) denotes the Banach space of all scalar Radon measures on X . It is proved that (B(\mathcal B o),\tau(B(\mathcal B o),M(X))) is a strongly Mackey space. For a sequentially complete locally convex Hausdorff space (E,\xi) , let M(X,E) denote the space of all Radon measures m:\mathcal B o\to E , equipped with the topology \mathcal T_s of setwise convergence. It is proved that a subset \mathcal R of M(X,E) is relatively \mathcal T_s -compact if and only if \mathcal R is uniformly regular and for each A\in\mathcal B o , the set \{m(A):m\in\mathcal R\} in E is relatively \xi -compact, if and only if the family \{T_m:m\in\mathcal R\} of corresponding integration operators T_m:B(\mathcal B o)\to E is (\tau(B(\mathcal B o),M(X)),\xi) -equicontinuous and for each v\in B(\mathcal B o) , the set \{\int_X v\,dm:m\in\mathcal R\} in E is relatively \xi -compact. As an application, we get a Nikodym theorem and a Dieudonné–Grothendieck type theorem on the setwise sequential convergence in M(X,E) .

Riesz-representable multilinear mappings

We introduce the class of Riesz-representable multilinear mappings on products of spaces for finite measures μ and give their main properties. We prove Grothendieck type theorems for composition with the natural operators and . Unlike the linear case, the composition is not nuclear but is characterized by what we call the left integral -factorization property.

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γW with supports in arbitrary subsets W of SpecR. If W is a specialization-closed subset, then γW coincides with the right derived functor RΓW of the section functor ΓW with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γW with W being an arbitrary subset.

A regularity principle in sequence spaces and applications

We prove a nonlinear regularity principle in sequence spaces which produces universal estimates for special series defined therein. Some consequences are obtained and, in particular, we establish new inclusion theorems for multiple summing operators. Of independent interest, we settle all Grothendieck's type (ℓ1,ℓ2) theorems for multiple summing multilinear operators. We further employ the new regularity principle to solve the classification problem concerning all pairs of admissible exponents in the anisotropic Hardy–Littlewood inequalities.

Random matrices and subexponential operator spaces

We introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces, as well as a C*-algebra that is subexponential with constant 1 but not exact. We also show that OH, R + C and max(l2) (or any other maximal operator space) are not subexponential.

Compactness and sequential completeness in some spaces of operators

Let $$X$$ be a completely regular Hausdorff space and $$C_b(X)$$ be the Banach lattice of all real-valued bounded continuous functions on $$X$$ , endowed with the strict topologies $$\beta _\sigma ,$$ $$\beta _\tau $$ and $$\beta _t$$ . Let $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ $$(z=\sigma ,\tau ,t)$$ stand for the space of all $$(\beta _z,\xi )$$ -continuous linear operators from $$C_b(X)$$ to a locally convex Hausdorff space $$(E,\xi ),$$ provided with the topology $$\mathcal{T}_s$$ of simple convergence. We characterize relative $$\mathcal{T}_s$$ -compactness in $$\mathcal{L}_{\beta _z,\xi }(C_b(X),E)$$ in terms of the representing Baire vector measures. It is shown that if $$(E,\xi )$$ is sequentially complete, then the spaces $$(\mathcal{L}_{\beta _z,\xi }(C_b(X),E),\mathcal{T}_s)$$ are sequentially complete whenever $$z=\sigma $$ ; $$z=\tau $$ and $$X$$ is paracompact; $$z=t$$ and $$X$$ is paracompact and Čech complete. Moreover, a Dieudonné–Grothendieck type theorem for operators on $$C_b(X)$$ is given.

Remarks on multiple summing operators on $$C(\Omega )$$ -spaces

We give the necessary and sufficient conditions for a multilinear bounded operator on \(C(\Omega _{1}) \times \cdots \times C(\Omega _{k}) \times X_{k+1}\times \cdots \times X_{k+n}\) to be multiple 1-summing. Based on this result we prove an inclusion result for multiple summing operators and an unexpected composition result of Grothendieck type for bilinear operators.

Operators on the space of bounded strongly measurable functions

Let L ( X , Y ) stand for the space of all bounded linear operators between real Banach spaces ( X , ‖ ⋅ ‖ X ) and ( Y , ‖ ⋅ ‖ Y ) , and let Σ be a σ-algebra of subsets of a non-empty set Ω. Let L ∞ ( Σ , X ) denote the Banach space of all bounded strongly Σ-measurable functions f : Ω → X equipped with the supremum norm ‖ ⋅ ‖ . A bounded linear operator T from L ∞ ( Σ , X ) to a Banach space Y is said to be σ-smooth if ‖ T ( f n ) ‖ Y → 0 whenever ‖ f n ( ω ) ‖ X → 0 for all ω ∈ Ω and sup n ‖ f n ‖ < ∞ . It is shown that if an operator measure m : Σ → L ( X , Y ) is variationally semi-regular (i.e., m ˜ ( A n ) → 0 as A n ↓ ∅ , where m ˜ ( A ) stands for the semivariation of m on A ∈ Σ ), then the corresponding integration operator T m : L ∞ ( Σ , X ) → Y is σ-smooth. Conversely, it is proved that every σ-smooth operator T : L ∞ ( Σ , X ) → Y admits an integral representation with respect to its representing operator measure. We prove a Banach–Steinhaus type theorem for σ-smooth operators from L ∞ ( Σ , X ) to Y. In particular, we study the topological properties of the space L ∞ ( Σ , X ) c ⁎ of all σ-smooth functionals on L ∞ ( Σ , X ) . We prove a form of a generalized Nikodým convergence theorem and characterize relative σ ( L ∞ ( Σ , X ) c ⁎ , L ∞ ( Σ , X ) ) -sequential compactness in L ∞ ( Σ , X ) c ⁎ . We derive a Grothendieck type theorem for L ∞ ( Σ , X ) c ⁎ . The relationships between different classes of linear operators on L ∞ ( Σ , X ) are established.

A Grothendieck-type theorem for the space of totally measurable functions

Let Σ be a σ-algebra of subsets of a non-empty set Ω. Let X be a real Banach space and let X * stand for the Banach dual of X. Let B(Σ, X) be the Banach space of Σ-totally measurable functions f: Ω → X, and let B(Σ, X) * and B(Σ, X) ** denote the Banach dual and the Banach bidual of B(Σ, X) respectively. Let bvca(Σ, X *) denote the Banach space of all countably additive vector measures ν: Σ → X * of bounded variation. We prove a form of generalized Vitali-Hahn-Saks theorem saying that relative σ(bvca(Σ, X *), B(Σ, X))-sequential compactness in bvca(Σ, X *) implies uniform countable additivity. We derive that if X reflexive, then every relatively σ( B(Σ, X) *, B(Σ, X))-sequentially compact subset of B(Σ, X) c ~ (= the σ-order continuous dual of B(Σ, X)) is relatively σ( B(Σ, X) *, B(Σ, X) **)-sequentially compact. As a consequence, we obtain a Grothendieck type theorem saying that σ( B(Σ, X) *, B(Σ, X))-convergent sequences in B(Σ, X) c ~ are σ( B(Σ, X) *, B(Σ, X) **)-convergent.

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.

The Euler structure and the Gysin homomorphism in oriented homology theories

The homological self-intersection formula, the Grothendieck type formula, and the excess formula are proved for oriented homology theories. Bibliography: 8 titles.

Hochschild cohomology of factors with property Γ

The main result of this paper is that the k th continuous Hochschild cohomology groups H k (M, M) and H k (M ,B (H)) of a von Neumann factor M⊆ B(H )o f type II 1 with property Γ are zero for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the �·� 2-norm of separately ultraweakly continuous multilinear maps, and combine these results to reduce to the case of completely bounded cohomology which is already solved.

Property gamma factors and the Hochschild cohomology problem.

The main result of this article is that the kth continuous Hochschild cohomology groups H(k)(M,M ) and H(k)(M, B(H)) of a von Neumann factor subset, dbl equals B(H) of type II(1) with property Gamma are 0 for all positive integers k. The method of proof involves the construction of hyperfinite subfactors with special properties and a new inequality of Grothendieck type for multilinear maps. We prove joint continuity in the parallel. parallel (2) norm of separately ultraweakly continuous multilinear maps and combine these results to reduce to the case of completely bounded cohomology, which is already solved.

Projectively bounded Fréchet measures

A scalar valued set function on a Cartesian product of σ \sigma -algebras is a Fréchet measure if it is a scalar measure independently in each coordinate. A basic question is considered: is it possible to construct products of Fréchet measures that are analogous to product measures in the classical theory? A Fréchet measure is said to be projectively bounded if it satisfies a Grothendieck type inequality. It is shown that feasibility of products of Fréchet measures is linked to the projective boundedness property. All Fréchet measures in a two dimensional framework are projectively bounded, while there exist Fréchet measures in dimensions greater than two that are projectively unbounded. A basic problem is considered: when is a Fréchet measure projectively bounded? Some characterizations are stated. Applications to harmonic and stochastic analysis are given.

Tensor Products and Grothendieck Type Inequalities of Operators in L p - Spaces

Tensor Products and Grothendieck Type Inequalities of Operators in L p - Spaces

Tensor products and Grothendieck type inequalities of operators in 𝐿_{𝑝}-spaces

Several results in the theory of ( p , q ) (p,q) -summing operators are improved by a unified but elementary tensor product concept.

Dilations of 𝑉-bounded stochastic processes indexed by a locally compact group

It is proved that a stochastic process (i.e., a Hilbert space valued function) indexed by a locally compact group is V V -bounded (i.e., weakly harmonizable in an appropriate sense) if, and only if, it can be expressed as an orthogonal projection of a process whose covariance function R R satisfies R ( s , t ) = ρ ( t − 1 s ) + ρ ( s t − 1 ) R(s,t) = \rho ({t^{ - 1}}s) + \rho (s{t^{ - 1}}) for some continuous positive-definite function ρ \rho . The result generalizes a well-known theorem due to H. Niemi, and depends on the noncommutative Grothendieck type inequality of G. Pisier.