The controlled separable projection property for Banach spaces

Let X, Y be two Banach spaces. We say that Y is a quasi-quotient of X if there is a continuous operator R: X → Y such that its range, R(X), is dense in Y.     

C(K) quotients of separable ℒ∞ spaces

It is shown that ifX is an ℒ⫗ space with separable dual, thenX has a quotient isomorphic toC(ω)^α if, and only if, there is anɛ >0, such that theɛ-Szlenk index ofX is at leastα. It was previously known, that ifX is a separable ℒ⫗ space with a non-separable dual, thenX has a quotient isomorphic toC[0, 1]. Research partially supported by NSF Grant MCS-74-07509-A03.     

On C(K) Grothendieck spaces

Using the theory of bounded linear operators, several characterizations of C(K) Grothendieck spaces are given. Supported by CDCH of ULA under project 1123-02-05-B 2000.     

A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

C(α) preserving operators on separable Banach spaces

Let T be a bounded linear operator from a separable Banach space X to a Banach space Y. A necessary and sufficient condition on T for the existence of a subspace Z of X such that Z is isomorphic to C(α) and the restriction of T to Z is an isomorphism is given. The special case where X is the disc algebra is then considered and results similar to those previously obtained by the author for C(K) spaces are obtained for the disc algebra. Finally some additional results of the same type are proved for subspaces of C(K) with small annihilator.     

On separable Lindenstrauss spaces

Isometric embeddings from l_∞ⁿin l_∞^{n + 1} can be described by a_i,n, i ? n, with $\text{∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{¦ a}{}_{\mathrm{i,n}}\text{¦ ? 1}$, such that e_i,n = e_{i,n + 1} + a_i,ne_{n + 1,n + 1}; i = 1,…, n; holds, where e_i,nand e_{i,n + 1} are the elements of the canonical unit vector bases of l_∞ⁿand l_∞^{n + 1}, respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a_i,n, i = 1,…, n, and the Banach space a_i,n, E_n ? E_{n + 1}, E_n ? l_∞ⁿ, where A determines the embeddings of the E_n. The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c₀and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C(K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A(S) where S is a Choquet simplex with dense extreme points.     

On the homotopy groups of separable metric spaces

The aim of this paper is to discuss the homotopy properties of locally well-behaved spaces. First, we state a nerve theorem. It gives sufficient conditions under which there is a weak n-equivalence between the nerve of a good cover and its underlying space. Then we conclude that for any (n−1)-connected, locally (n−1)-connected compact metric space X which is also n-semilocally simply connected, the nth homotopy group of X, πn(X), is finitely presented. This result allows us to provide a new proof for a generalization of Shelah?s theorem (Shelah, 1988 [18]) to higher homotopy groups (Ghane and Hamed, 2009 [8]). Also, we clarify the relationship between two homotopy properties of a topological space X, the property of being n-homotopically Hausdorff and the property of being n-semilocally simply connected. Further, we give a way to recognize a nullhomotopic 2-loop in 2-dimensional spaces. This result will involve the concept of generalized dendrite which introduce here. Finally, we prove that each 2-loop is homotopic to a reduced 2-loop.     

On extreme points in separable conjugate spaces

It is proved that every bounded closed and convex subset of an arbitrary conjugate separable Banach space is the closed convex hull of its extreme points.     

On the relationship between cmp and def for separable metrizable spaces

For each pair of positive integers k and m with k?m there exists a separable metrizable space X(k,m) such that cmpX(k,m)=k and defX(k,m)=m. This solves Problem 6 from [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993, p. 71].     

On quasi-Souslin C c (X) spaces

We prove that the space C _c (X) of the real-valued continuous functions with the compact-open topology is quasi-Souslin iff it is K-analytic. This implies that C _c (X) is K-analytic iff it is dominated by the irrationals. Partially supported by the project MTM2005-01182 of the Spanish Ministry of Education and Science, co-financed by the European Community (Feder projects).     

On the necessity of property (α) for some vector-valued multiplier theorems

The notion of R-bounded operator families has proven to be crucial to several aspects of vector-valued Harmonic Analysis and its applications to PDEs. Concrete and important R-bounded sets have been identified among operators acting on UMD Banach spaces, which in addition enjoy G. Pisier’s property (α). We prove the necessity of (α) for a number of conclusions of this kind, complementing analogous results of G. Lancien in the product-space theory of Fourier multipliers and H ^∞-calculus. Received: 24 January 2007 Revised: 1 June 2007     

On the space of all regular operators from C(K) into C(K)

It is known that L^r(E, C(K)), the space of all regular operators from E into C(K), is a Riesz space for all Riesz spaces E if and only if K is Stonian. We prove that this statement holds if E is replaced by C(K), where K is a compact space, the cardinal number of which satisfies a certain condition.     

On shadowing property for inverse limit spaces

We study here the G-shadowing property of the shift map σ on the inverse limit space X _f, generated by an equivariant self-map f on a metric G-space X.      

SeparableG spaces are isomorphic toC(K) spaces

We show that a wide class of separable preduals ofL ¹(μ) spaces, namely, theG spaces, introduced by Grothendieck, are isomorphic toC(K) spaces. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

On the non-uniqueness of minimal projection in spaces

The main objective of this note is to exhibit a simple example of subspaces UL_p(μ) (p≠2) that admit two different projections with minimal norm. While for p=1,∞, such subspaces are well-known [W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1<p<∞ their existence was open.     

On the approximating local lifting property for operator spaces

In this paper, we define and study the approximately local lifting property for operator spaces. We show that an operator space V has the approximately local lifting property if and only if V^∗ is injective. This implies that an operator space V has the approximately local lifting property if and only if it has the local lifting property.     

On a separable weak version of the bounded approximation property

Recently, Lee introduced and studied the separable weak bounded approximation property (BAP). Lee proved that the separable weak BAP of \({X^*}\), the dual space of a Banach space \({X}\), coincides with the BAP of \({X^*}\) whenever \({X^{**}}\) has the weak Radon–Nikodým property. We show that the separable weak BAP and the BAP are always the same properties.     

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.