Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $N$ be a von Neumann algebra, and $A$ be a dense $*$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that:

For any $x\in N^+$, there exists a increasing net $(x_j)$ in $A^+$ such that $x_j \to x$ in the ultraweak topology ?

The case of $A$ being an ideal of $N$ seems known (it is right?) but my question is about the general case.

share|improve this question

1 Answer 1

The following might be a counterexample. Let $N=L^{\infty}([0,1])$ and $A=C([0,1])$, the subalgebra of continuous functions. Take $x\in N\setminus \{0\}$ to be a nonnegative function such that the set $\{t\in [0,1]:x(t)=0\}$ has positive measure in every subinterval. The only element $f\in A^+$ such that $f\le x$ is $f\equiv 0$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.