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 sense? why is it allowed to divide by the positive element? I saw such expression in an article, I wa...

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 wasn't able of doing much more. I wa...

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?

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 inner product $\langle ., . \rangle$ iff $\lVert \...

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 elements $x,y\in \overline{\bigcup_n L_n}$ with $x\leqslant y$, can we find two sequences $(x_n)...