For questions about vector lattices (aka. Riesz spaces; a type of vector space equipped with a partial order), Banach lattices and similar topics.
16
votes
1answer
151 views
Properties of the Cone of Positive Semidefinite Matrices
The set of positive semidefinite symmetric real matrices form a cone. We can define an order over the set of matrices by saying $X\geq Y$ if and only if $X-Y$ is positive semidefinite. I suspect that ...
1
vote
0answers
10 views
2
votes
1answer
73 views
Planar cross section of leech lattice?
I was wondering what any planar cross sections of the leech lattice would look like. I don't know much about this topic at all, I'm just quite curious. Is there any way to find out?
0
votes
1answer
74 views
How to `bound' $L^\infty$ by the constant function $1$
Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is ...
3
votes
0answers
85 views
A question regarding vector spaces with partial order
$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$
Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
2
votes
0answers
46 views
Are order continuous Banach lattices with weak unit weakly sequentially complete?
Let $E$ be an order continuous Banach lattice with a weak unit represented as $L^\infty(\mu) \subseteq E\subseteq L^1(\mu)$. Is it true that $E$ is also a weakly sequentially complete Banach lattice?
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2
votes
1answer
86 views
completeness of cones in an ordered normed space
Let $(X,\|\cdot\|,\le)$ be a normed, ordered vector space over $\mathbb{R}$ and let $X^+=\lbrace x\in X:x\ge0\rbrace$ denote the (positive) cone in $X$. with a metric $d$ induced by the norm ...
5
votes
1answer
180 views
Show sequence equicontinuous
I don't know how to prove this question:
Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
6
votes
1answer
223 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?
3
votes
1answer
186 views
How does positivity affect operator norms?
Take a measure space $\Omega$, an exponent $1 \le p < \infty$ and a linear operator $T \colon L^p(\Omega) \to L^p(\Omega)$. Suppose that $T$ is positive, that is $Tf \ge 0$ a.e. if $f \ge 0$ almost ...
