Schauder decompositions and the Fremlin projective tensor product of Banach lattices

In this paper, we first introduce a lattice decomposition and finite-dimensional lattice decomposition (FDLD) for Banach lattices. Then we show that for a Banach lattice with FDLD, the following are equivalent: (i) it has the Radon-Nikodym property; (ii) it is a KB-space; (iii) it is a Levi space; and (iv) it is a σ-Levi space. We then give a sequential representation of the Fremlin projective tensor product of an atomic Banach lattice with a Banach lattice. Using this sequential representation, we show that if one of the Banach lattices X and Y is atomic, then the Fremlin projective tensor product has the Radon-Nikodym property (or, respectively, is a KB-space) if and only if both X and Y have the Radon-Nikodym property (or, respectively, are KB-spaces).     

Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

Banach格上的b-Dunford-Pettis算子

为了进一步研究Banach格上算子的性质,受b-序有界集和Dunford-Pettis集定义的启发,给出了b-Dunford-Pettis算子的定义,研究了该算子与b-AM-紧算子(Dunford-Pettis全连续算子,弱极限算子,序Dunford-Pettis算子)间的关系;利用b-Dunford-Pettis算子与Dunford-Pettis算子的共轭关系,证明了b-Dunford-Pettis算子满足控制性.     

The complete continuity properties for the positive projective tensor product of atomic Banach lattices

Let E be an atomic Banach lattice and X be a separable Banach lattice. In this paper we show that ${E \hat{\otimes}_{|\pi|} X,}$ the positive projective tensor product of E and X, has the complete continuity property (respectively, the analytic complete continuity property) if and only if both E and X have the same property. We also discuss the inheritance of type I- and type II-complete continuity properties by ${E \hat{\otimes}_{|\pi|} X}$ from E and X.     

Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism

Let A and B be Banach algebras and T : B → A be a continuous homomorphism. nweak amenability of the Banach algebra A ×T B(defined in Bade, W. G., Curtis, P. C., Dales, H. G.:Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc.,55(2), 359–377(1987)) is studied. The new version of a Banach algebra defined with a continuous homomorphism is introduced and Arens regularity and various notions of amenability of this algebra are studied.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

Nonlinear Carleman operators on Banach lattices

An operator, not necessarily linear, will be called a Carleman operator if the image of the positive elements in the unit ball are bounded in the universal completion of the range space. For certain Banach lattices, a class of (not necessarily linear) Carleman operators is characterized in terms of an integral representation and in a more general setting as operators satisfying a pointwise finiteness condition. These operators though not linear are orthogonally additive and monotone.

     

On irreducible operators on Banach lattices

We prove an extension of Ando-Krieger's theorem for positive, irreducible, order continuous Harris operators on Dedekind complete Banach lattices.     

Quadratic stochastic operators on Banach lattices

We study the convergence of iterates of quadratic stochastic operators that are mean monotonic. They are defined on the convex set of probability measures concentrated on a weakly compact order interval \(S = [0, f]\) of a fixed Banach lattice F. We study their regularity and identify the limits of trajectories either as the “infimum” or “supremum” of the support of initial distributions.     

自反子空间格的表示和运算

证明了von Neumann 代数的子空间格的自反性和KS- 性都不依赖于该von Neumann 代数的正规忠实*- 表示; 引入了von Neumann 代数及所含子空间格的半自由积运算, 证明了两子空间格的半自由积同构于它们的直和.