What would you do if you see a notion which you used to work with has two names and the same name you use for this notion is used for another notion too. For example, after the work of J.B. Bost, A. Connes, (Hecke algebras, type III factors and phase transition with spontaneous symmetry breaking in number theory. Selecta Math. (New Series) Vol.1, no.3, 411-456, (1995)), in operator algebras community people call a subgroup $H$ of a group $G$ almost normal if every double coset of $H$, like $HgH$ is the union of finitely left cosets. It is equivalent to say that $H$ is almost normal if it is commensurable with its conjugates. Therefore in group theory people call such a subgroup conjugate commensurable. In stead, in group theory people call a subgroup $H$ of a group $G$ almost normal if its normalizer is of finite index in $G$. Now if I use the word conjugate commensurable for the above notion it is confusing in operator algebras community and if I use almost normal it can be confused with group theorists' notion of almost normal subgroups. What is the best way to deal with these sort of problems.



When a name is used for two different notions!
To start with the obvious, in a given work you should conform to the usage that the majority of your readers expect. A second principle is that you should work out some reasonable system for your own work and follow that consistently. In cases where you expect readers from both camps, you will need to frequently draw your readers' attention to the meaning of your notation.


This is not a problem you can solve, you just have to find a solution you can live with. And whatever you find, your referees will disagree with you (and possibly with each other).


