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Register a new userPersistence of Banach lattices under nonlinear order isomorphisms
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Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection between them that preserves order. We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to c_0, and for Banach lattices that contain a closed sublattice order isomorphic to c_0.
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Let $X$ be a topological space. A subset of $C(X)$, the space of continuous real-valued functions on $X$, is a partially ordered set in the pointwise order. Suppose that $X$ and $Y$ are topological spaces, and $A(X)$ and $A(Y)$ are subsets of $C(X)$ and $C(Y)$ respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between $A(X)$ and $A(Y)$. Under some general assumptions on $A(X)$ and $A(Y)$, and when $X$ and $Y$ are compact Hausdorff, it is shown that existence of an order isomorphism between $A(X)$ and $A(Y)$ gives rise to an associated homeomorphism between $X$ and $Y$. This generalizes a classical result of Kaplansky concerning linear order isomorphisms between $C(X)$ and $C(Y)$ for compact Hausdorff $X$ and $Y$. The class of near vector lattices is introduced in order to extend the result further to noncompact spaces $X$ and $Y$. The main applications lie in the case when $X$ and $Y$ are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, local and bounded local
The classical theorems of Banach and Stone, Gelfand and Kolmogorov, and Kaplansky show that a compact Hausdorff space $X$ is uniquely determined by the linear isometric structure, the algebraic structure, and the lattice structure, respectively, of the space $C(X)$. In this paper, it is shown that for rather general subspaces $A(X)$ and $A(Y)$ of $C(X)$ and $C(Y)$ respectively, any linear bijection $T: A(X) to A(Y)$ such that $f geq 0$ if and only if $Tf geq 0$ gives rise to a homeomorphism $h: X to Y$ with which $T$ can be represented as a weighted composition operator. The three classical results mentioned above can be derived as corollaries. Generalizations to noncompact spaces and other function spaces such as spaces of uniformly continuous functions, Lipschitz functions and differentiable functions are presented.
[REVISED VERSION] The aim of this paper is to state a sharp version of the Konig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.
We prove that every isometry between two combinatorial spaces is determined by a permutation of the canonical unit basis combined with a change of signs. As a consequence, we show that in the case of Schreier spaces, all the isometries are given by a change of signs of the elements of the basis. Our results hold for both the real and the complex cases.
Associated to a nonzero homomorphism $varphi$ of a Banach algebra $A$, we regard special functionals, say $m_varphi$, on certain subspaces of $A^ast$ which provide equivalent statements to the existence of a bounded right approximate identity in the corresponding maximal ideal in $A$. For instance, applying a fixed point theorem yields an equivalent statement to the existence of a $m_varphi$ on $A^ast$; and, in addition we expatiate the case that if a functional $m_varphi$ is unique, then $m_varphi$ belongs to the topological center of the bidual algebra $A^{astast}$. An example of a function algebra, surprisingly, contradicts a conjecture that a Banach algebra $A$ is amenable if $A$ is $varphi$-amenable in every character $varphi$ and if functionals $m_varphi$ associated to the characters $varphi$ are uniformly bounded. Aforementioned are also elaborated on the direct sum of two given Banach algebras.
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