The phrase "on trust" implies a sense of reliance and faith in someone or something, and there are several synonyms that can be used to convey this same notion. "In good faith" is a common phrase that describes a situation where one party is trusting another without any doubts or reservations. Another alternative is "with confidence," which indicates a feeling of trust and belief in someone's abilities or intentions. "Depend on" and "count on" are also synonyms that can be used to indicate trust or reliability. Ultimately, the choice of synonym will depend on the context and tone of the situation, but they all convey the same sense of trust and faith.