Loyalty and betrayal are linked to The Tempest's larger themes of servitude and freedom; either feeling is motivated by how each individual perceives his position relative to others. Antonio's betrayal of his brother and theft of the dukedom of Milan are the source of conflict in the play, but the action contemporary to the play follows a series of attempted betrayals. Alonso and Prospero would both be murdered by traitors, but this is thwarted by the actions of loyal characters like Ariel and Gonzalo. Loyalty and treachery also serve as the two main personality traits of the players. You can separate the loyal out, and divide the bad into those who were misguided and now repentant, and those who are just plain evil.
Loyalty is a farce in the play; everyone follows the courtly rule of swearing loyalty, but gives up on the notion as soon as it is no longer convenient.
Antonio's betrayal of Prospero reminds us that even family members cannot be counted on to be loyal.