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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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$s$-convexity implies $p$-convexity in Banach lattices

If $1<p<s<\infty$ and $E$ is a Banach lattice which is $s$-convex, is it also $p$-convex? If so, what would be a good reference for this?
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25 views

Bounded set and norm bounded set in a Banach lattice space

I am reading about Banach lattice space and confuse a little bit about two concepts "bounded" and "norm bounded set". Could you please help me to declare them? More precisely, let $E$ be a Banach ...
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Proof of additivity of the positive linear functional $\phi^+$ on a vector/Banach lattice that will be $\phi\vee0$.

For context, this is used in defining $\phi\vee0$ in a proof that the dual of a vector lattice is a vector lattice. Given a linear functional $\phi$ on a vector lattice $V$, define $\phi^+$ on $V^+$ ...
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29 views

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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44 views

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 ...
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14 views

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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11 views

$(V, \geq)$ Banach lattice, $(W, \geq)$ Riesz space, then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$

Let $(V, \geq)$ be a Banach lattice and $(W, \geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$, where ...
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26 views

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 ...
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2answers
70 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 ...
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1answer
40 views

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(...
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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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1answer
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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 $C[0,1]$. ...
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123 views

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 ...
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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 ...
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1answer
106 views

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 ...
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208 views

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 $Q$ ...
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60 views

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 ...
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1answer
57 views

“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-...
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207 views

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 ...
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1answer
58 views

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 ...
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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,...
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196 views

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 ...
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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 $E^{**}$ ...
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1answer
109 views

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 $...
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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 ...
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1answer
109 views

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 ...
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1answer
636 views

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?