Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip(X,E) and Lip(Y,F), for strictly convex normed spaces E and F and metric spaces X and Y:

(i)

Characterize those base spaces X and Y for which all isometries are weighted composition maps.

(ii)

Give a condition independent of base spaces under which all isometries are weighted composition maps.

(iii)

Provide the general form of an isometry, both when it is a weighted composition map and when it is not.

In particular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.     

Extreme points of Banach lattices related to conditional expectations

Let (X,F,μ) be a complete probability space, B a sub-σ-algebra, and Φ the probabilistic conditional expectation operator determined by B. Let K be the Banach lattice {f∈L¹(X,F,μ):‖Φ(|f|)_‖∞<∞} with the norm ‖f‖=‖Φ(|f|)_‖∞. We prove the following theorems:

(1)

The closed unit ball of K contains an extreme point if and only if there is a localizing set E for B such that supp(Φ(χE))=X.

(2)

Suppose that there is n∈N such that f?nΦ(f) for all positive f in L^∞(X,F,μ). Then K has the uniformly λ-property and every element f in the complex K with is a convex combination of at most 2n extreme points in the closed unit ball of K.

     

A conjecture of Palamodov about the functors Ext in the category of locally convex spaces

In 1971 Palamodov proved that in the category of locally convex spaces the derived functors Ext^k(E,X) of Hom(E,·) all vanish if E is a (DF)-space, X is a Fréchet space, and one of them is nuclear. He conjectured a “dual result”, namely that Ext^k(E,X)=0 for all if E is a metrizable locally convex space, X is a complete (DF)-space, and one of them is nuclear. Assuming the continuum hypothesis we give a complete answer to this conjecture: If X is an infinite-dimensional nuclear (DF)-space, then

(1)

There is a normed space E such that Ext¹(E,X)≠0.

(2)

where is a countable product of lines.

(3)

Ext^k(E,X)=0 for all k?3 and all locally convex spaces E.

     

Quasi-Suslin weak duals

Cascales, Ka?kol, and Saxon (CKS) ushered Kaplansky and Valdivia into the grand setting of Cascales/Orihuela spaces E by proving:

(K)

If E is countably tight, then so is the weak space(E,σ(E,E^′)), and

(V)

(E,σ(E,E^′))is countably tight iff weak dual(E^′,σ(E^′,E))is K-analytic.

The ensuing flow of quasi-Suslin weak duals that are not K-analytic, a la Valdivia's example, continues here, where we argue that locally convex spaces E with quasi-Suslin weak duals are (K, V)'s best setting: largest by far, optimal vis-a-vis Valdivia. The vaunted CKS setting proves not larger, in fact, than Kaplansky's. We refine and exploit the quasi-LB strong dual interplay.     

The Fréchet-Urysohn property of Pixley-Roy hyperspaces

Let F[X] be the Pixley-Roy hyperspace of a regular space X. In this paper, we prove the following theorem.

Theorem. For a space X, the following are equivalent:

(1)

F[X]is a k-space;

(2)

F[X]is sequential;

(3)

F[X]is Fréchet-Urysohn;

(4)

Every finite power of X is Fréchet-Urysohn for finite sets;

(5)

Every finite power ofF[X]is Fréchet-Urysohn for finite sets.

As an application, we improve a metrization theorem onF[X].     

Optimum quantization and its applications

Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Tóth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d?2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on

(i)

the minimum distortion of high-resolution vector quantization and optimum quantizers,

(ii)

the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures,

(iii)

the minimum error of numerical integration formulae for classes of Hölder continuous functions and optimum sets of nodes,

(iv)

best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and

(v)

the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes.

     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

•

T contains all weakly Lindelöf Banach spaces;

•

l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

•

T is stable under weak homeomorphisms;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

•

E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

Function spaces from core compact coherent spaces to continuous B-domains

It is proved in this paper that for a continuous B-domain L, the function space [X→L] is continuous for each core compact and coherent space X. Further, applications are given. It is proved that:

(1)

the function space from the unit interval to any bifinite domain which is not an L-domain is not Lawson compact;

(2)

the Isbell and Scott topologies on [X→L] agree for each continuous B-domain L and core compact coherent space X.

     

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for

(i)

bilinear forms over an algebraically closed or real closed field;

(ii)

sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and

(iii)

sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F; the canonical matrices are based on any given set of canonical matrices for similarity over F.

A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (1988) 481-501].     

Strong zero-dimensionality of hyperspaces

For a space X, X² denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K(X) denotes the collection of all non-empty compact sets of X with the subspace topology of X². The following are known:

•

ω² is not normal, where ω denotes the discrete space of countably infinite cardinality.

•

For every non-zero ordinal γ with the usual order topology, K(γ) is normal iff whenever cf γ is uncountable.

In this paper, we will prove:

(1)

ω² is strongly zero-dimensional.

(2)

K(γ) is strongly zero-dimensional, for every non-zero ordinal γ.

In (2), we use the technique of elementary submodels.     

Intersection cohomology of the circle actions

A classical result says that a free action of the circle S¹ on a topological space X is geometrically classified by the orbit space B and by a cohomological class e∈H²(B,Z), the Euler class. When the action is not free we have a difficult open question:

(Π)

“Is the space X determined by the orbit space B and the Euler class?”

The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space B and the Euler class determine:

•

the intersection cohomology of X,

•

the real homotopy type of X.

     

Banach spaces with the Daugavet property, and the centralizer

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties:

(1)

Every member of R has the Daugavet property.

(2)

It Y is a member of R, then, for every Banach space X, both the space L(X,Y) (of all bounded linear operators from X to Y) and the complete injective tensor product lie in R.

(3)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C(K,(Y,τ)) (of all Y-valued τ-continuous functions on K) is a member of R.

(4)

If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C(K,Y)-superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195-206]) are members of R.

(5)

All dual Banach spaces without minimal M-summands are members of R.

     

Both compact and sequentially compact sets in abelian topological group

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,

(1)

if G is a Hausdorff abelian topological group and μ:N²→G is countably additive, then the range μ(N²)={μ(A):A⊆N} is compact metrizable;

(2)

if X is a Hausdorff locally convex space and {xj}⊂X, then F={_∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X^′〉-polar topology having the same subseries convergent series as the weak topology.

     

Normality and countable paracompactness of hyperspaces of ordinals

For an ordinal α, α² denotes the collection of all nonempty closed sets of α with the Vietoris topology and K(α) denotes the collection of all nonempty compact sets of α with the subspace topology of α². It is well known that α² is normal iff cfα=1. In this paper, we will prove that for every nonzero-ordinal α:

(1)

α² is countably paracompact iff cfα≠ω.

(2)

K(α) is countably paracompact.

(3)

K(α) is normal iff, if cfα is uncountable, then cfα=α.

In (3), we use elementary submodel techniques.     

Spaces with large weak extent

In this paper, we show the following statements:

(1)

For any cardinal κ, there exists a pseudocompact centered-Lindelöf Tychonoff space X such that we(X)?κ.

(2)

Assuming ℵ₀²=ℵ₁², there exists a centered-Lindelöf normal space X such that we(X)?ω₁.

     

Computing tight upper bounds on the algebraic connectivity of certain graphs

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let B be a generalized Bethe tree. The algebraic connectivity of:

the generalized Bethe tree B,

a tree obtained from the union of B and a tree T isomorphic to a subtree of B such that the root vertex of T is the root vertex of B,

a tree obtained from the union of r generalized Bethe trees joined at their respective root vertices,

a graph obtained from the cycle C_r by attaching B, by its root, to each vertex of the cycle, and

a tree obtained from the path P_r by attaching B, by its root, to each vertex of the path,

is the smallest eigenvalue of a special type of symmetric tridiagonal matrices. In this paper, we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices. Finally, we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs.

     

Sequential properties of function spaces with the compact-open topology

The main results of the paper are:

(1)

If X is metrizable but not locally compact topological space, then Ck(X) contains a closed copy of S₂, and hence does not have the property AP;

(2)

For any zero-dimensional Polish X, the space Ck(X,2) is sequential if and only if X is either locally compact or the derived set X^′ is compact; and

(3)

All spaces of the form Ck(X,2), where X is a non-locally compact Polish space whose derived set is compact, are homeomorphic, and have the topology determined by an increasing sequence of Cantor subspaces, the nth one nowhere dense in the (n+1)st.

     

Continuity of posets via Scott topology and sobrification

In this paper, posets which may not be dcpos are considered. The concept of embedded bases for posets is introduced. Characterizations of continuity of posets in terms of embedded bases and Scott topology are given. The main results are:

(1)

A poset is continuous iff it is an embedded basis for a dcpo up to an isomorphism;

(2)

A poset is continuous iff its Scott topology is completely distributive;

(3)

A topological T₀ space is a continuous poset equipped with the Scott topology in the specialization order iff its topology is completely distributive and coarser than or equal to the Scott topology;

(4)

A topological T₁ space is a discrete space iff its topology is completely distributive.

These results generalize the relevant results obtained by J.D. Lawson for dcpos.