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?
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2$\begingroup$ Well, you could define $f$ and $g$ to have a ratio "to the right" if there exists a unique element $a$ such that $ag = f$, and in this case define $f/g := a$. Similarly, you could define a ratio "to the left" $g\backslash f$. But this seems to work for general algebras (or rings, or even semigroups...), so it doesn't seem to have much to do with von Neumann algebras. I'm not sure, though, if you can maybe characterize the existence of such ratios $f/g$ and $g\backslash f$ in the von Neumann algebra setting (say, for instance, in terms of a spectral condition). $\endgroup$– Jochen GlueckCommented Sep 16, 2019 at 20:19
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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, original article). (The adaptation to von Neumann algebras is not terribly difficult, and can be found, for example, in $\S$3.4.2 of my thesis.)
