For functions spaces we have ratio of two functions namley $\frac{f}{g}$, Can ratio of two operators in von Neumann algebra make sense? If at all it makes sense, what will be the proper definition of it?



On analogue of ratio in operator algebras
Let $a$ and $b$ be elements of a von Neumann algebra $\mathscr{A}$. As Jochen Glueck notes, one needs to differentiate between $a/b$ and $a\backslash b$. I would define $a/b$ to be (provided it exists) the unique element $c$ of $\mathscr{A}$ with $a=cb$ and $\mathop{Ran}(b)^\perp \subseteq \mathop{Ker}(c)$. Such a $c$ exists if and only if $a^\*a\leq N b^\*b$ for some natural number $N$. For Hilbert spaces I believe this is know as Douglas' Lemma ([wikipedia](http:///https://en.wikipedia.org/wiki/Douglas%), [original article](https://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0203464-1/S0002-9939-1966-0203464-1.pdf)). (The adaptation to von Neumann algebras is not terribly difficult, and can be found, for example, in $\S$3.4.2 of [my thesis](https://arxiv.org/pdf/1804.02203.pdf).)


