严格循环的算子代数

设X是半自共轭的Banach空间,而A是B(X)的严格循环的交换子代数,考虑算子代数A的一些性质,这些性质与算子代数的自反性有关,而且这些性质为讨论A的自反性提供了理论基础.     

An elementary proof of the principle of local reflexivity

We give an elementary proof of the principle of local reflexivity.

     

On the Duals of Segre Varieties

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M.M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?     

Lifting of the approximation property from Banach spaces to their dual spaces

Inspired by the principle of local reflexivity, due to Lindenstrauss and Rosenthal, a new geometric property of Banach spaces, the extendable local reflexivity, was recently introduced by Rosenthal. Johnson and Oikhberg proved that the extendable local reflexivity permits lifting the bounded approximation property from Banach spaces to their dual spaces. It is not known whether the extendable local reflexivity permits lifting the approximation property. We prove that it does whenever the space is complemented in its bidual. This involves the concept of the weak bounded approximation property, introduced by Lima and Oja.

     

Maluta's coefficient in Musielak-Orlicz sequence spaces equipped with the Orlicz norm

Maluta's coefficient of Musielak-Orlicz sequence spaces equipped with the Orlicz norm is calculated. A sufficient condition for the Schur property of these spaces is given.

     

Multipliers of weighted spaces and reflexivity property

We prove for some translation-invariant weighted spaces the following characterization: is a multiplier of if and only if leaves invariant every translation-invariant subspace of . This result is equivalent with the reflexivity of the multiplier algebra of .

     

A Weak Property β Equals Reflexivity for a Banach Space

We introduce a weak property () in Aerms of the De Blasi measure of weak noncompactness and prove that this property characterizes the reflexivity.     

A characterization of reflexive Banach spaces

A Banach space is not reflexive if and only if there exist a closed separable subspace of and a convex closed subset of with empty interior which contains translates of all compact sets in . If, moreover, is separable, then it is possible to put .

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Uniformly non-l n (1) Musielak-Orlicz sequence spaces

We give a necessary and sufficient condition for the uniformly non-l _n ⁽¹⁾ property of Musielak-Orlicz sequence spacesl ^Φ generated by a sequence Φ=(ϕ_n:n⩾l) of finite Orlicz functions such that for eachn∈ℕ. As a result, forn ₀⩾2, there exist spacesl ^Φ which are only uniformly non-l _n ⁽¹⁾ forn⩾n ₀. Moreover we obtain a characterization of uniformly non-l _n ⁽¹⁾ and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l _n ⁽¹⁾ Nakano sequence spaces. This extends a result of Luxemburg in [19]. Submitted in memory of Professor W. Orlicz