The lattice copies of ?∞(Γ)/c0(Γ) in a quotient of a Banach lattice

Let E be a Dedekind σ-complete Banach lattice, let E_a denote the order continuous part of E, andsuppose that E_a is order dense in E. If E contains a lattice copy of ?_∞ (equivalently, E ≠ _a) then E/E_a contains a lattice copy of ?_∞/c₀, and hence a lattice-isometric copy of ?_∞. This is one of the consequences of more general results presented in this paper.     

The approximation property for spaces of holomorphic functions on infinite dimensional spaces II

Let H(U) denote the vector space of all complex-valued holomorphic functions on an open subset U of a Banach space E. Let τω and τδ respectively denote the compact-ported topology and the bornological topology on H(U). We show that if E is a Banach space with a shrinking Schauder basis, and with the property that every continuous polynomial on E is weakly continuous on bounded sets, then (H(U),τω) and (H(U),τδ) have the approximation property for every open subset U of E. The classical space c₀, the original Tsirelson space T^∗ and the Tsirelson^∗-James space are examples of Banach spaces which satisfy the hypotheses of our main result. Our results are actually valid for Riemann domains.     

Dunford-Pettis property

The following theorem is proved. If a locally convex space, quasi-complete for Mackey topology, has D-P (Dunford-Pettis) property then it has strict D-P property. Conversely, if (E′, σ(E′, E)) has a σ-compact dense subset and E has strict D-P property, then it has D-P property. Also it is proved that (C_b(X),$\text{F}$) where F=β₀, β, orβ₁, has strict D-P property and (C_b(X), β₀) has D-P property; if X contains a σ-compact dense subset then (C_b(X), β) and (C_b(X), β₁) have D-P property.     

Weighted composition operators on nonlocally convex weighted spaces of continuous functions

LetV be a system of weights on a completely regular Hausdorff spaceX and letB(E) be the topological vector space of all continuous linear operators on a general topological vector spaceE. LetCV ₀(X, E) andCV _b (X, E) be the weighted spaces of vector-valued continuous functions (vanishing at infinity or bounded, respectively) which are not necessarily locally convex. In the present paper, we characterize in this general setting the weighted composition operatorsW _π,? onCV ₀(X, E) (orCV _b (X, E)) induced by the operator-valued mappings π:X→B(E) (or the vector-valued mappings π:X→E, whereE is a topological algebra) and the self-map ? ofX. Also, we characterize the mappings π:X→B(E) (or π:x→E) and ?:X→X which induce the compact weighted composition operators on these weighted spaces of continuous functions.     

Topologies on products of partially ordered sets III: Order convergence and order topology

This paper deals with the question under which circumstances filter-theoretical order convergence in a product of posets may be computed componentwise, and the same problem is treated for convergence in the order topology (which may differ from order convergence). The main results are:

1. Order convergence in a product of posets is obtained componentwise if and only if the number of non-bounded posets occurring in this product is finite (1.5).
2. For any product of posets, the projections are open and continuous with respect to the order topologies (2.1).
3. A productL of chainsL _i has topological order convergence iff all but a finite number of the chains are bounded. In this case, the order topology onL agrees with the product topology (2.7).
4. If (L _i :j ∈J) is a countable family of lattices with topological order convergence and first countable order topologies then order topology of the product lattice and product topology coincide (2.8).
5. LetP ₁ be a poset with topological order convergence and locally compact order topology. Then for any posetP ₂, the order topology ofP ₁?P ₂ coincides with the product topology (2.10).
6. A latticeL which is a topological lattice in its order topology is join- and meet-continuous. The converse holds whenever the order topology ofL?L is the product topology (2.15).

Many examples are presented in order to illustrate how far the obtained results are as sharp as possible.     

Complementation and decompositions in some weakly Lindelöf Banach spaces

Let Γ denote an uncountable set. We consider the questions if a Banach space X of the form C(K) of a given class (1) has a complemented copy of c₀(Γ) or (2) for every c₀(Γ)⊆X has a complemented c₀(E) for an uncountable E⊆Γ or (3) has a decomposition X=A⊕B where both A and B are nonseparable. The results concern a superclass of the class of nonmetrizable Eberlein compacts, namely Ks such that C(K) is Lindelöf in the weak topology and we restrict our attention to Ks scattered of countable height. We show that the answers to all these questions for these C(K)s depend on additional combinatorial axioms which are independent of ZFC ± CH. If we assume the P-ideal dichotomy, for every c₀(Γ)⊆C(K) there is a complemented c₀(E) for an uncountable E⊆Γ, which yields the positive answer to the remaining questions. If we assume ♣, then we construct a nonseparable weakly Lindelöf C(K) for K of height ω+1 where every operator is of the form cI+S for c∈R and S with separable range and conclude from this that there are no decompositions as above which yields the negative answer to all the above questions. Since, in the case of a scattered compact K, the weak topology on C(K) and the pointwise convergence topology coincide on bounded sets, and so the Lindelöf properties of these two topologies are equivalent, many results concern also the space Cp(K).     

Chain conditions and weak topologies

We study conditions on Banach spaces close to separability. We say that a topological space is pcc if every point-finite family of open subsets of the space is countable. For a Banach space E, we say that E is weakly pcc if E, equipped with the weak topology, is pcc, and we also consider a weaker property: we say that E is half-pcc if every point-finite family consisting of half-spaces of E is countable. We show that E is half-pcc if, and only if, every bounded linear map E→c₀(ω₁) has separable range. We exhibit a variety of mild conditions which imply separability of a half-pcc Banach space. For a Banach space C(K), we also consider the pcc-property of the topology of pointwise convergence, and we note that the space Cp(K) may be pcc even when C(K) fails to be weakly pcc. We note that this does not happen when K is scattered, and we provide the following example:

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There exists a non-metrizable scattered compact Hausdorff space K with C(K) weakly pcc.

     

Generalizations of methods of best approximation to convex optimization in locally convex spaces. II: Hyperplane theorems

We show that for a continuous convex functional f on a locally convex space E and a convex subset G of E such that inf f(E) < inf f(G), the problems of computing inf f(G) and of characterizing the elements g₀?G with f(g₀) = inf f(G) can be reduced to the same problems for a suitable hyperplane H₀ in E.     

Dini-Verbände mit fast gut frisiertem Dualkegel

Summary We prove a theorem which is a generalisation of a theorem due to Goullet de Rugy: LetE be a locally convex vector lattice such that the structure spaces of the principal ideals ofE are universally measurable in their Stone-ech-compactifications respectively. Then, ifE is infrabarrelled or its topology is finer than the topology of compact convergence,E is a Dini lattice iffE ₊ is nearly well-capped.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

A note on monolithic scattered compacta

For a Banach space E, it is well-known that a necessary condition for E to have the controlled separable complementation property (CSCP  , for short) is that the dual unit ball _BE?$B_{E^{?}}$ be monolithic in the weak-star topology. We prove here that when X is a scattered first countable locally compact space, then monolithicity of X   turns out to be sufficient for _C0(X)$C_0(X)$ to enjoy the CSCP.     

Continuity and equicontinuity of semigroups on norming dual pairs

We study continuity and equicontinuity of semigroups on norming dual pairs with respect to topologies defined in terms of the duality. In particular, we address the question whether continuity of a semigroup already implies (local/quasi) equicontinuity. We apply our results to transition semigroups and show that, under suitable hypothesis on E, every transition semigroup on C _b (E) which is continuous with respect to the strict topology β ₀ is automatically quasi-equicontinuous with respect to that topology. We also give several characterizations of β ₀-continuous semigroups on C _b (E) and provide a convenient condition for the transition semigroup of a Banach space valued Markov process to be β ₀-continuous.     

A lattice approach to narrow operators

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L₁ is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N_r(E, F) of all narrow regular operators is a band in the vector lattice L_r(E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N_r(E, F) in L_r(E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : E → F from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T_D + T_N where T_D is a sum of an order absolutely summable family of disjointness preserving operators and T_N is narrow. Supported by Ukr. Derzh. Tema N 0103Y001103.     

Certain locally convex topologies on the discrete vector-valued Lebesgue spaces

Let Γ be a set and (E, ‖·‖ _E ) be a nontrivial Banach space. In this paper, through generalizing to vector-valued discrete Lebesgue spaces ? ¹(Γ,E), we show that the topology β ¹(Γ,E) introduced by Singh is, in fact, a type of strict topology. This observation enables us to conclude various basic properties of β ¹(Γ,E). Then, we consider the discrete semigroup algebra ? ¹(S,E) under certain locally convex topologies. As an application of our results, we show that the semigroup algebra (? ¹(S,E), β ¹(S,E)) with the convolution as multiplication is a complete semi-topological (but not topological) algebra.     

Resolvent estimates for a certain class of Schrödinger operators with exploding potentials

Let $\text{H = ?Δ + V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)+ V(x)}$ be a Schrödinger operator in Rⁿ. Here $\text{V}{}_{\mathrm{<mtext>E</mtext>}}\text{(¦x¦)}$ is an “exploding” radially symmetric potential which is at least C² monotone nonincreasing and O(r²) as r → ∞. V is a general potential which is short range with respect to V_E. In particular, V_E  0 leads to the “classical” short-range case (V being an Agmon potential). Let Λ = lim_{r → ∞}V_E(r) and R(z) = (H ? z)^?1, 0 < Im z, Λ < Re z < ∞. It is shown that R(z) can be extended continuously to Im z = 0, except possibly for a discrete subset $\text{N}$?(Λ, ∞), in a suitable operator topology $\text{B(L, L}{}^1\text{)}$. And L ? L²(Rⁿ) is a weighted L²-space; H is then absolutely continuous over (Λ, ∞), except possibly for a discrete set of eigenvalues. The corresponding eigenfunctions are shown to be rapidly decreasing.     

The Banach-Stone theorem revisited

Let X and Y be compact Hausdorff spaces, and E and F be locally solid Riesz spaces. If π:C(X,E)→C(Y,F) is a 1-biseparating Riesz isomorphism then X and Y are homeomorphic, and E and F are Riesz isomorphic. This generalizes the main results of [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829] and [X. Miao, C. Xinhe, H. Jiling, Banach-Stone theorems and Riesz algebras, J. Math. Anal. Appl. 313 (1) (2006) 177-183], and answers a conjecture in [Z. Ercan, S. Önal, Banach-Stone theorem for Banach lattice valued continuous functions, Proc. Amer. Math. Soc. 135 (9) (2007) 2827-2829].     

On countable bounded tightness for spaces Cp(X)

It is well known that the space C_p([0,1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space C_p(X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if C_p(X) has countable bounded tightness, i.e., for every subset A of C_p(X) and every x in the closure of A in C_p(X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.