I was reading about these two algorithms, but I don't understand how they can be used in a recommender system because using the MovieLens dataset these algorithms recommend the best movie for all the users and not for a specifc user. I mean does multi armed bandit do the same predictions for all users?
A/B testing is not a recommendation algorithm per se, but a method used to evaluate and compare systems such as, for example, recommender systems.
You would use A/B testing in a live setting, where a recommender system is used to provide recommendations to users. By assigning users to two groups and giving both of these groups recommendations coming from two different systems (usually one representing the status quo and the other representing a new variant), you can compare both groups and determine if the new variant leads to better results than the status quo.
The name "A/B testing" usually refers to this kind of analysis performed on a live setting. If you are doing offline analysis, on a dataset like MovieLens, then you wouldn't usually call it A/B testing.
Methods for multi-armed bandits address a different problem: balancing exploration and exploitation.
The idea is that once you begin using a recommender system, the content your users consume is influenced by the output of this recommender system. This is a problem, because data about these consumption patterns is usually used to train the recommender system, which means the system would be trained on data influenced by its own predictions.
The most basic example of a multi-armed bandits approach would be to use your recommender system on half the users, give random recommendations to the other half, and use only the data from the latter half's behaviour to train the recommender system.
Because multi-armed bandit systems act by controlling how training data is collected, it doesn't make much sense to talk about them in the context of a dataset that has already been collected, like MovieLens.