Yet on linear structures of norm-attaining functionals on Asplund spaces

In this paper, we show that if an Asplund space X is either a Banach lattice or a quotient space of C(K), then it can be equivalently renormed so that the set of norm-attaining functionals contains an infinite dimensional closed subspace of X^* if and only if X^* contains an infinite dimensional reflexive subspace, which gives a partial answer to a question of Bandyopadhyay and Godefroy.     

Metric Projections and Polyhedral Spaces

We prove that if X is a Banach space and Y is a proximinal subspace of finite codimension in X such that the finite dimensional annihilator of Y is polyhedral, then the metric projection from X onto Y is lower Hausdorff semi continuous. In particular this implies that if X and Y are as above, with the unit sphere of the annihilator space of Y contained in the set of quasi-polyhedral points of X ^*, then the metric projection onto Y is Hausdorff metric continuous. Partially supported under project DST/INT/US-NSF/RPO/141/2003.     

Hyperbolic imbedding and spaces of continuous extensions of holomorphic maps

LetX be a complex subspace of a complex spaceY. We show that hyperbolic imbeddedness ofX inY is characterized by relative compactness in the compact-open topology of certain spaces of continuous extensions of holomorphic maps from the punctured diskD* toX and fromM -A toX whereM is a complex manifold andA is a divisor onM with normal crossings. We apply these characterizations to obtain generalizations and extensions of theorems of Kobayashi, Kiernan, Kwack, Noguchi and Vitali forD and for higher dimensions. Relative compactness ofX inY is not assumed.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

A characterization of maximal non-k-factor-critical graphs

A graph G of order p is k-factor-critical,where p and k are positive integers with the same parity, if the deletion of any set of k vertices results in a graph with a perfect matching. G is called maximal non-k-factor-critical if G is not k-factor-critical but G+e is k-factor-critical for every missing edge e∉E(G). A connected graph G with a perfect matching on 2n vertices is k-extendable, for 1?k?n-1, if for every matching M of size k in G there is a perfect matching in G containing all edges of M. G is called maximal non-k-extendable if G is not k-extendable but G+e is k-extendable for every missing edge e∉E(G) . A connected bipartite graph G with a bipartitioning set (X,Y) such that |X|=|Y|=n is maximal non-k-extendable bipartite if G is not k-extendable but G+xy is k-extendable for any edge xy∉E(G) with x∈X and y∈Y. A complete characterization of maximal non-k-factor-critical graphs, maximal non-k-extendable graphs and maximal non-k-extendable bipartite graphs is given.     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

Embedding of a pseudo-point residual design into a Möbius plane

Let $\text{U}$ be a class of subsets of a finite set X. Elements of $\text{U}$ are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, $\text{U}$) is called a (υ, k, λ) t-design, denoted by S_λ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in $\text{U}$, |A| = k. A Möbius plane M is an S₁(3, q+1, q²+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M^*. It can be easily checked that M^* is an S_q(2, q+1, q²). Any S_q(2, q+1, q²) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M^*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M^* is said to have the Tangency Property if for any block A in M^*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M^* is uniquely embeddable into a Möbius plane if and only if M^* satisfies the Tangency Property.     

On factors ofC([0, 1]) with non-separable dual

LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT ^* X ^* non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC. The research for this paper was partially supported by NSF-GP-30798X. An erratum to this article is available at .     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

Neighborhoods of sphere-like continua

Let X be an S^k-like continuum in Euclicean space Eⁿ. It is shown that if the embedding of X satisfies the small loops condition and 1⩽k⩽n−4, then X has arbitrarily small neighborhoods which are homeomorphic to S^k×B^n−k. It follows that if X and Y are S^k-like continua in Sⁿ, 1⩽k⩽n−4, and both satisfy the small loops condition, then X and Y have homeomorphic complements if and only if they have the same shape.     

A note on reflexivity in Banach spaces

For any normed spaceX, the unit ball ofX is weak *-dense in the unit ball ofX ^**. This says that for any ε>0, for anyF in the unit ball ofX ^**, and for anyf ₁,…,f _n inX ^*, the system of inequalities |f _i(x)?F(f _i)|≤ε can be solved for somex in the unit ball ofX. The author shows that the requirement that ε be strictly positive can be dropped only ifX is reflexive.     

Some sets obeying harmonic synthesis

LetX be a (not necessarily closed) subspace of the dual spaceB ^* of a separable Banach spaceB. LetX ₁ denote the set of all weak ^* limits of sequences inX. DefineX _a , for every ordinal numbera, by the inductive rule:X _a = (U _{b < a} X _b ) ₁ .There is always a countable ordinala such thatX _a is the weak ^* closure ofX; the first sucha is called theorder ofX inB ^* . LetE be a closed subset of a locally compact abelian group. LetPM(E) be the set of pseudomeasures, andM(E) the set of measures, whose supports are contained inE. The setE obeys synthesis if and only ifM(E) is weak ^* dense inPM(E). Varopoulos constructed an example in which the order ofM(E) is 2. The authors construct, for every countable ordinala, a setE inR that obeys synthesis, and such that the order ofM(E) inPM(E) isa. This work was done in Jerusalem, when the second-named author was a visitor at the Institute of Mathematics of the Hebrew University of Jerusalem, with the support of an NSF International Travel Grant and of NSF Grant GP33583.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

Asymptotically isometric copies of c0 and l in certain Banach spaces

If X is a separable Banach space, then X∗ contains an asymptotically isometric copy of l¹ if and only if there exists a quotient space of X which is asymptotically isometric to c₀. If X is an infinite-dimensional normed linear space and Y is any Banach space containing an asymptotically isometric copy of c₀, then L(X,Y) contains an isometric copy of l^∞. If X and Y are two infinite-dimensional Banach spaces and Y contains an asymptotically isometric copy of c₀, then contains a complemented asymptotically isometric copy of c₀.     

Strong Dunford-Pettis sets and spaces of operators (Monatsh. Math. 144, 275–284 (2005))

In a recent paper, Ghenciu and Lewis studied strong Dunford-Pettis sets and made the following two assertions:

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

Strong Dunford-Pettis sets and spaces of operators (Monatsh. Math. 144, 275–284 (2005))

In a recent paper, Ghenciu and Lewis studied strong Dunford-Pettis sets and made the following two assertions:

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

On J. H. C. Whitehead's aspherical question I

A connected, finite two-dimensional CW-complex with fundamental group isomorphic toG is called a [G, 2] _f -complex. LetL⊲G be a normal subgroup ofG. L has weightk if and only ifk is the smallest integer such that there exists {l ₁,…,l _k}⊆L such thatL is the normal closure inG of {l ₁,…,l _k}. We prove that a [G, 2] _f -complexX may be embedded as a subcomplex of an aspherical complexY=X∪{e ₁ ² ,…,e _k ² } if and only ifG has a normal subgroupL of weightk such thatH=G/L is at most two-dimensional and defG=defH+k. Also, ifX is anon-aspherical [G, 2] _f -subcomplex of an aspherical 2-complex, then there exists a non-trivial superperfect normal subgroupP such thatG/P has cohomological dimension ≤2. In this case, any torsion inG must be inP.