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Questions tagged [banach-lattices]

Banach lattices are Banach spaces endowed with a partial ordering that is compatible with the norm.

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Closed subspaces of $L^1(\mathbb{R})$ that is not isomorphic to a subspace of a space with an unconditional basis

It is known that $L^1([0,1])$ does not admit any unconditional basis. Even more, $L^1([0,1])$ is not isomorphic to a subspace of a space with an unconditional basis. My question is the following ``...
Roddick Yu's user avatar
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Positivity of the bidual operator

How is the fixed space of a bounded linear operator related to the fixed space of the bidual operator? Motivation of question: I have a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach ...
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$\lambda$-sum of $L^1$ is a Banach space

I have some troubles in proof that $\lambda$-sum of $L_1$ is a banach space, is mensioned in a Wnuk paper that is a banach space but i dont understand how can i proof that is complete. Let $\lambda \...
Juan Rubio's user avatar
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Can a Banach space and its dual both contain copies of $\ell_1$?

Each AL space $E$ contains a copy of $\ell_1$. Its dual $E'$ is an AM space. Is it possible for $E'$ to contain a copy of $\ell_1$ as well?
HardyHulley's user avatar
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Norm inequality for Banach Lattices

Let $X$ be a Banach lattice, $\varepsilon>0$ and $x,y\in S_X$. If $\||x|\pm y\|\le1+\varepsilon$, then is it true that $\||x|+|y|\|\le1+f(\varepsilon)$? Where $f$ is some real function satisfying $\...
Stefano Ciaci's user avatar
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2 answers
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Kernel operators on AL-spaces

Let $E$ be an AL-space. For simplicity $E=L_{1}(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a strictly localizable measure space. Let $T:E\rightarrow E$ be a bounded kernel operator on $E$ with ...
user44155's user avatar
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Is every AM space a $C^*$-algebra?

A Banach lattice $E$ is said to be an AM-space if $$\|\sup\{x,y\}\|=\sup\{\|x\|,\|y\|\}$$ for all positive $x,y\in E$. My question is as follows: Is every AM-space (which is a $*$-algebra) a $C^*$-...
modeltheory's user avatar
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Eigenvectors of the dual of positive irreducible operators

Let $E$ be a Banach Lattice such that $E$ is a $M$-space. Assume that $T:E\to E$ is a positive bounded non-compact irreducible linear operator with positive spectral radius. And define \begin{align*} ...
Matheus Manzatto's user avatar
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Pre-dual of a lattice is a lattice?

Let $X$ be a Banach space such that its dual $X'$ is a Banach lattice. Then must $X$ also be a lattice? I know that if $X$ is a lattice, then so is its dual $X'$. However, I was wondering is the ...
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Dual of a order continuous Banach Lattice $\subseteq L^1$

I have recently read a theorem in a paper (porbably) indicating that if we take a Banach Lattice $E \subseteq L^1$ with order continuous norm, then we can identify its dual $E'$ with another space $F \...
PsychoRhano's user avatar
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Is the space $W^{1,1}_0(0,1)$ an abstract $L^1$ space?

Let $W^{1,1}_0(0,1)$ be the space of functions on the interval $(0,1)$ that vanish at the boundary with the standard $W^{1,1}(0,1)$-norm. $$||f|| = \int_0^1 |f(x)| \, dx + \int_0^1 |f'(x)| \, dx.$$ ...
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Positivity in $C^*$-algebra vs Riesz spaces

If $A$ is a $C^*$-algebra, then a self-adjoint element $x\in A$ is a called positive if $sp(x)\subseteq [0,\infty)$. I know of the following result: There exist positive elements $x_+,x_-\in A$ such ...
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Extending a linear operator satisfying an order condition

Let $\ell^\infty$ be the usual space of bounded sequences, and consider the subspace $V_1 ⊂ \ell^\infty$ consisting of vectors with finite $1$-norm. (That is, $V_1$ contains those $x ∈ \ell^\infty$ ...
Jeff Russell's user avatar
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Lattice of self-adjoint bounded functionals on C*-algebra

When it comes to real-valued signed measures on a measurable space $(X, \mathcal{B})$, there’s a lattice structure given by \begin{align*} (\mu \lor \nu) (A) & = \sup \{ \mu (A \cap B) + \nu (A \...
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Banach Lattices and the Fatou property: can I always find disjoint positive elements with large norms?

Suppose I have a lattice $X$ with a weak Fatou norm, meaning that there exists some constant $M$ (clearly $\geq 1$) such that for all $x\geq 0$ and for all increasing nets $x_\alpha$ with $x_\alpha \...
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Quasi-interior points of a sublattice

Let $1\leq p< \infty$ and $E= L^p(\Omega,\mu)$. If $A \subseteq \Omega$ is a measurable set and $T: E \to E$ is defined by $Tf=\mathbf{1}_Af$, then find the quasi-interior points of $\operatorname{...
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Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
Mark Roelands's user avatar
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General form of elements in the vector lattice (Riesz Space) generated by a vector space

Can any one tell me how the elements of the set $A^{∧∨}$ will look like? I believe they are of the following form: $∨^m_i∧^{m_i}_{k_i} x_{i,{k_i}}$ where $x_{i,{k_i}}$ is in $A$. But in this case I'm ...
Prince Khan's user avatar
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Is an abelian $C^*-$ algebra an $AM-$ space?

First, I remark some definitions: Given $E$ a normed vector lattice, if the norm on $E$ satisfies \begin{equation*} \|x \vee y\| = sup(\|x\|,\|y\|) \quad (x,y \in E_+), \end{equation*} then $(E,\|\...
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Banach lattice from a positive linear form on the dual

I'm trying to understand the following construction: Let $E$ a Banach lattice. Basically we want to construct for each $\nu \in E^*$ (A positive linear functional $\nu:E \rightarrow \mathbb{C})$, a ...
James Arten's user avatar
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Examples of Normalized Semigroup of Operators

In the theory of semigroups of bounded linear operators ${Z(t)}_{t\geq0}$, the normalized semigroups, $Z(t) : V\rightarrow V$ are defined as follows: $$Z(t)e=e,$$ where $V$ is unital Banach Algebra ...
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Positiveness of Inverse Of Positive Operator Implies Lattice Isomorphism?

Let $X$ be a Banach Lattice and denote by $\mathcal{B}(X)$ the banach space of bounded linear endomorphisms. An operator $T \in \mathcal{B}(X)$ is called positive if $Tx \geq 0$ whenever $x \geq 0.$ ...
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Product of two weakly compact endomorphisms is compact

I've seen the statement that if a Banach lattice $E$ satisfies the property that $$||x + y|| = ||x|| + ||y||, \: \forall x,y \in E_+$$ then if $S,T:E \to E$ are weakly compact endomorphisms, then $ST$...
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Weakly null sequence in Banach lattices

Let $(x_n)_n$ be a positive, disjoint, weakly null sequence in a Banach lattice $E$. If $(y_n)_n$ is a sequence such that $0\leq y_n\leq x_n$ for every $n\in \mathbb{N}$, we can garantee that $y_n$ is ...
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closed lattice ideal is isomorphic to $C(K)$

Let $X\in E$, where $E$ is a Banach function space on $(0,1)$. Consider the interval $[-X,X]$ and generate it to a closed lattice ideal $I$ of $E$. We may renorm this ideal $I$ such that $[-X,X]$ is ...
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Establish Archimedean property of a vector-lattice

I am trying to find ways to prove the Archimedian property of a certain vector lattice and got stuck on the following type of problem. I feel the statement below (or in fact weaker versions) should ...
makkiato's user avatar
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Unbounded norm convergence

I study on unbounded norm convergence in Banach lattice. I have to find an example.Is there any example of unbounded norm convergence but not norm convergence in Banach lattice?
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Reference for Banach Lattices

I am looking for a survey or a basic book on Banach Lattices (I know next to nothing about the subject, so a basic survey is what I believe will be helpful). Can somebody please suggest any good ...
Carl Butcher's user avatar
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2 answers
259 views

Infinite dimensional Banach lattice $L^\infty(X)$ is not order continuous

Consider an arbitrary measure space $(X,\Sigma,\mu)$, with the only assumption being that $L^\infty(X)$ is infinite dimensional. Consider $L^\infty(X)$ as a Banach lattice with the usual ordering. As ...
K.Power's user avatar
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Complexification of a Banach lattice

If $E$ is a real Banach lattice, then complexification of $E$ is defined as follows: $$E_c= E+iE=\{x+iy:x,y \in E\}\\(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2)\\(x_1+iy_1)(x_2+iy_2)=(x_1x_2-y_1y_2)+i(...
Sahiba Arora's user avatar
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Irreducible operators

I would like some reference to books that talk about irreducible operators on Banach lattices and its properties. A quick google search did not reap fruitful results. Edit 1: I'm especially ...
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Questions about $Lip_k[0,1]$

Let $Lip_k[0,1] = \{f \in C([0,1]) : |f(x)-f(y)| \leq k*|x-y|\}$. Prove or disprove the following statements. $Lip_k[0,1] $ is a subalgebra of $C[0,1]$. $Lip_k[0,1] $ is a sublattice of $C[0,1]$. ...
TuringTester69's user avatar
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Closed subalgebras of $C[0,1]$ and $C[0,1)$?

Let $C[0,1]$ be the algebra of continuous functions $f: [0,1] \to \mathbb R$. Let $C[0,1)$ be the algebra of bounded continuous functions $f: [0,1) \to \mathbb R$. Can anyone give some interesting ...
Daron's user avatar
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$x'_n\overset{w^\ast}{\to}0$ and $x_n\overset{w}{\to}0$ implies $\sup_m |x_m'|(|x_n|)\to 0$?

If $x_n'$ is weak-$\ast$-ly null and $x_n$ is weakly null in a Banach lattice, do we always have $$\sup_m |x_m'|(|x_n|)\to 0\,?$$ Thank you! Background: Definition weak(-$\ast$) topology: On the ...
Carucel's user avatar
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Hardy space as a Banach lattice

The Hardy spaces $H^p$ of holomorphic functions on the unit disk are Banach spaces. Question: Are they also Banach lattices? If yes, why is it less common to consider the Hardy spaces as Banach ...
Carucel's user avatar
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Show that two projections commute if and only if their respectively closed subspaces are compatibles

I have the next problem: Let $\mathcal{H}$ be a separable and complex Hilbert space, with $S$ and $Q$ two closed subspaces on it. Let $P_S$ and $P_Q$ be the orthogonal projection onto $S$ and $Q$ ...
Nehuila's user avatar
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Strong convergence of regular operators and convergence of the modulus

Let $E$ be a Banach lattice and $T_n\in\mathcal{L}^r(E)$ a sequence of regular operators such that $T_n$ converges strongly to $T\in\mathcal{L}^r(E)$. How to prove that $\left|T_n\right|$ converges ...
metic's user avatar
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"Representation Capacity" of Finite Lattice Ordered Modules

Apologies for what is probably a very basic question, but I am looking for references for what sort of lattices can be represented in what I think should be called something like "finite lattice-...
lvilnis's user avatar
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1 answer
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Positive cone of Banach lattice algebra

From the literature on Banach lattice algebras (and that on ordered Banach algebras) there does not appear to be a consensus on a definition. What is agreed is that one should be a Banach lattice, be ...
Shinning Star's user avatar
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Division in Banach Lattice Algebra

Let X be a Banach Lattice Algebra and $X_+=\{f\in X: f>0\}$. Let $f:X_+\rightarrow X$ be continuously differentiable. Question: When does the expression $\frac{f'(x)}{x}$ for $x\in X_+$, make ...
Shinning Star's user avatar
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Is a closed subset of a Banach lattice complete?

Let $(E, || \cdot||)$ be a Banach lattice. Let $E_{+}$ denote the positive cone of $E$. The metric $\rho$ on $E_{+}$ is induced by the complete lattice norm $|| \cdot ||$, which is defined by $\rho(x,...
Paradiesvogel's user avatar
9 votes
1 answer
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Banach space that is not a Banach lattice

There is a well-known criterium that distinguishes Banach spaces into the following two classes: those Banach spaces that can be made into a Hilbert space $(X, \langle .,. \rangle)$ and those that ...
yada's user avatar
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4 votes
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Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$ ...
S -'s user avatar
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1 vote
1 answer
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Separation Hahn Banach theorem in vector Banach lattice

Let $X$ be a vector Banach lattice. Let $C$ be a closed cone of positive elements in $X^+$ and let $0\leq x\in X-C$. Q: Does there exists any bounded positive linear functional $f$ on $X$ by which $...
ABB's user avatar
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A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
Jest's user avatar
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3 votes
1 answer
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Are dual spaces with unconditional bases weakly sequentially complete?

It is well known that a weakly sequentially complete Banach space with an unconditional basis is isomorphic to a conjugate space. Is the converse to this statement true? If a Banach space is a ...
Jimmy's user avatar
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12 votes
1 answer
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C*-algebras as Banach lattices?

It seems to be trivial but I am not sure about monotonicity of the norm in the non-commutative case: Is every C*-algebra a Banach lattice with respect to its natural positive cone?
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