Describe the trust Prospero had for Antonio.

Prospero's trust in his brother, Antonio, is the reason that Prospero and Miranda are stranded on their island in Shakespeare's The Tempest.
Prospero, the rightful Duke of Milan, entrusted Antonio with ever-increasing responsibility for administering the day-to-day affairs of Prospero's dukedom. Thus, Antonio began to believe that he, not Prospero, was (or should be) the actual Duke of Milan. Ultimately, this caused Antonio to betray Prospero's trust in him.
The word "trust" appears only twice in the play, both times spoken by Prospero within four lines of each other. Prospero wants Miranda to understand (and Shakespeare wants his audience to understand) the limitless trust that Prospero placed in Antonio and the depth of Antonio's betrayal of that trust.

PROSPERO: . . . and my trust,Like a good parent, did beget of himA falsehood, in its contrary as greatAs my trust was, which had indeed no limit . . . (1.2.110–113)

Antonio conspired with the King of Naples to overthrow Prospero and seize his dukedom. Antonio caused Prospero and Miranda to be cast adrift to die in a leaky excuse for a boat that even the rats had deserted, but somehow, Prospero and Miranda managed to make their way to the remote island that has been their home for the past twelve years.
Even more than that, Prospero's misplaced trust in his brother is the reason for the play itself. If Prospero had not trusted Antonio so explicitly, and if Antonio had not betrayed that trust so completely, Prospero and Miranda would not have been cast adrift at sea. They would never have come to this island; the ship bearing Antonio and the King of Naples would not have been passing by Prospero's island; the tempest which nearly destroyed their ship would never have occurred (or had any reason to occur); and Prospero would not have been able to right the grievous wrongs committed against him by his brother twelve years ago.