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Questions tagged [banach-lattices]

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

A Banach lattice is a complete normed vector lattice such that $|x| \leq |y| \implies \|x\| \leq \|y\|$.

For example, $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the usual norm and order given by $(x_1,x_2,\ldots,x_n)\leq (y_1,y_2,\ldots y_n) \iff x_i\leq y_i$ for every $1\leq i\leq n.$ Other examples include $L^p(\Omega) (1\leq p\leq \infty), C(K) (K:$compact), $C_0(\Omega) (\Omega:$ locally Hausdorff).

47 questions
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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?
Batiscaf
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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 cannot. A norm $\lVert . \rVert$ is induced by an…
yada
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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 $x ∈ \ell^\infty$ such that $\sum_i |x_i|$ is finite. I'm not calling…
Jeff Russell
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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 $Q$ respectively. We say that $S$ and $Q$ are…
Nehuila
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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 an associative algebra with a sub-multiplicative…
Shinning Star
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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 $\rho(x,y) := || x - y||$ for all $x,y \in E_{+}$. My…
Paradiesvogel
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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 $E^{**}$ I have been able to prove that $1) \implies 2)$ but I…
S -
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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 $C^*$-algebra? My intuition is as follows: I think of…
modeltheory
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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 an exercise in Operator Theoretic Aspects of…
K.Power
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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 conjugate space and has an unconditional basis, must it…
Jimmy
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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. $$||f|| = \int_0^1 |f(x)| \, dx + \int_0^1 |f'(x)| \, dx.$$ Is this space a Banach lattice? It appears to…
doc
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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 that $x=x_+-x_-$ and $x_+x_-=0$. Now I…
Guest
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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 $x_\alpha \uparrow x$, we have $ \| x \| \leq M \| x_\alpha…
user5644262
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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 $ST$ is compact. I have yet to see a proof of this…
Debreu
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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 interested in properties of the adjoint of an irreducible…
Mark
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