Does Caesar trust Antony in act one of Julius Caesar?

In Act One of Julius Caesar, Caesar does not seem to distrust Antony. He openly shares his thoughts and plans with Antony and sends him on important missions. In the first scene of the play, he tells Antony to "Let me have men about me that are fat, / Sleek-headed men, and such as sleep o' nights." He also asks him to "observe well who they be that bid me tremble, / For, in the base degrees, great men have thriven." This indicates that Caesar trusts Antony's judgment and loyalty. Later in the scene, when Cassius tries to persuade Brutus to join the conspiracy against Caesar, he mentions Antony as one of Caesar's closest friends and potential successors. This suggests that Antony is widely seen as someone close to Caesar and trusted by him.