Its possible to retain a Fragment between Activities?
Lets say I have Activity A with Fragment F_Left placed at the left and F
Based on your response to my comment, I have a slightly different answer. It may not end up being the best answer in your specific situation, I'll let you decide that. :)
Right now you are bundling your fragments in activities because that is what made sense to you, but really, you can probably treat the entire process as one activity and use fragment transactions to hide & show (or create and destroy) fragments as needed.
Since you won't be creating and destroying activities, your menu fragment on the left will be left untouched, and you won't have any problems with its UI state. The set of operations you want to run (which no doubt includes all sorts of different fragments on the right) does not need to be launched in a new activity - but you will have to find a way to manage the logic you need for the fragment transactions (either in your one über-activity or in some kind of OperationsManager class).
I think this will end up being a lot smoother for the users of your application since the single activity just remains running - and you are only changing the parts that actually need to change.