On the controlled separable projection property for some C(K) spaces

We give necessary and sufficient conditions for a compact K in order to guarantee that C(K) enjoys the controlled separable projection property (CSPP). As a consequence, we obtain an example of a scattered compact K such that C(K) is not injected into a Hilbert space, although C(K)* is, and nevertheless C(K) does not have the CSPP.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

LFC bumps on separable Banach spaces

In this note we construct a C^∞-smooth, LFC (Locally depending on Finitely many Coordinates) bump function, in every separable Banach space admitting a continuous, LFC bump function.     

Frame expansions in separable Banach spaces

Banach frames are defined by straightforward generalization of (Hilbert space) frames. We characterize Banach frames (and Xd-frames) in separable Banach spaces, and relate them to series expansions in Banach spaces. In particular, our results show that we can not expect Banach frames to share all the nice properties of frames in Hilbert spaces.     

Multivalued differential equations in separable Banach spaces

This paper is concerned with multivalued differential equations of the form F(t,x), whereF is a multivalued mapping taking as its values nonempty compact, but not necessarily convex, subsets in a separable Banach space. The main result is connected with the existence of solutions of these equations.     

Nonresonance problems for differential inclusions in separable Banach spaces

Let be a real separable Banach space. The boundary value problem

is studied on the infinite interval Here, the closed and densely defined linear operator generates an evolution operator The function is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either is bounded and is compact for or is compact and is equicontinuous. The mapping is a bounded linear operator and is fixed. The nonresonance problem is solved by using Ma's fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

     

Existence of separable uniformly homeomorphic nonisomorphic Banach spaces

An example is given proving that there exist two separable Banach spaces which are uniformly homeomorphic but not isomorphic.     

Operator-decomposability of Gaussian measures on separable Banach spaces

An operator-decomposable Gaussian measure on a separable Banach space can be factorized into a convolution product of a strongly operator-decomposable Gaussian measure and an operator-invariant Gaussian measure (with respect to the same operator). An example for this very factorization is discussed in some detail. In particular it is shown that a strongly operator-decomposable Gaussian measure need not necessarily be supported by the contraction subspace of the operator involved. Finally, the decomposability semigroup of a Gaussian measure turns out to be convex; and the corresponding invariance semigroup belongs to its extreme boundary.     

Non-dentable sets in Banach spaces with separable dual

A non-RNP Banach spaceE is constructed such thatE ^* is separable and the RNP is equivalent to the PCP on the subsets ofE.     

Characterization of Gaussian semigroups on separable Banach spaces

Let E be a separable real Banach space and denote by BUC (E) the space of bounded and uniformly continuous functions on E. For a C ₀-semigroup acting on BUC(E), we obtain necessary and sufficient conditions ensuring that is Gaussian.     

Ball-covering property of Banach spaces

We consider the following question: For a Banach spaceX, how many closed balls not containing the origin can cover the sphere of the unit ball? This paper shows that: (1) IfX is smooth and with dimX=n<∞, in particular,X=R ⁿ,then the sphere can be covered byn+1 balls andn+1 is the smallest number of balls forming such a covering. (2) Let Λ be the set of all numbersr>0 satisfying: the unit sphere of every Banach spaceX admitting a ball-covering consisting of countably many balls not containing the origin with radii at mostr impliesX is separable. Then the exact upper bound of Λ is 1 and it cannot be attained. (3) IfX is a Gateaux differentiability space or a locally uniformly convex space, then the unit sphere admits such a countable ball-covering if and only ifX ^* isw ^*-separable.     

Smooth extensions of functions on separable Banach spaces

Let X be a Banach space with a separable dual X*. Let ${Y\subset X}Let X be a Banach space with a separable dual X*. Let Y ì X{Y\subset X} be a closed subspace, and f:Y?\mathbbR{f:Y\rightarrow\mathbb{R}} a C ¹-smooth function. Then we show there is a C ¹ extension of f to X.     

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗̌_ɛ Y be the injective tensor product of X and Y.

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.     

A double-dual characterization of separable Banach spaces containingl 1

It is proved that a separable Banach spaceB contains a subspace isomorphic tol ¹ if (and only if) there exists an element inB**, the double-dual ofB, which is not a weak* limit of a sequence of elements inB. ConsequentlyB contains an isomorph ofl ¹ if (and only if) the cardinality ofB** is greater than that of the continuum. The research of the second-named author was partially supported by NSF — GP — 30798X1     

Asymmetry and projection constants of Banach spaces

We discuss various asymmetry constants of finite-dimensional Banach spaces in a more generalized frame than that of [2], and solve a problem raised in [7] by finding an increasing sequence of Banach spaces whose diagonal asymmetry constants tend to infinity. We investigate the question of whether the projection constant of everyn-dimensional Banach space is strictly less than , and show that this is so whenn=2. The research for this paper was partially supported by NSF-GP-34193.