Is the business owner trusted to tell me that the place is kosher? In other words, should I believe him?

There are no halakhic rules about this, because with prohibitions we are not dealing with testimony but with an impression. Therefore, you can ask him specifically rather than settling for a general statement that it is kosher, and if you get the impression that he is telling the truth, then believe him.

If he himself does not keep kosher for his own purposes, then the Shulchan Arukh already ruled:
One who is suspected of eating forbidden foods—whether he is suspected regarding a Torah prohibition or a rabbinic prohibition—one may not rely on him concerning them, and if one is hosted by him, one should not eat from his food in matters regarding which he is suspected.

As I wrote, there are no rules here, and therefore it is impossible to issue a Jewish-law ruling on this. It is a question of trustworthiness and persuasion, not a formally halakhic question. The rule that one witness is trusted regarding prohibitions is not testimony in the accepted halakhic sense.

The rule that one witness is trusted regarding prohibitions is not testimony in the usual sense, but here too there is a rule that someone suspected regarding a matter is not trusted about it. And that makes sense, because even if the person is not a liar, there is a high chance that he is unaware of or not paying attention to important details, or forgot, or cut corners. Even in places with a kosher certificate, there are often significant kosher problems. All the more so here.

All those considerations are indeed sensible, and each person can decide for himself whether he is convinced or not. But that is not because there is such a halakhic rule.

By the way, there are several proofs that the rule that someone suspected regarding a matter may not testify about it does not stem from suspicion of lying, but is rather a formal rule in the laws of testimony.

This is not very far from the case of meat left out of sight. If he himself does not keep kosher and there is no one supervising, how do you know whether he does not occasionally let friends hold an event in the restaurant with food brought from outside, or let workers from the neighboring office take a meal to the office and return the utensils afterward, after using the non-kosher office microwave? These are everyday occurrences. So even if there is no rule about testimony by someone under suspicion, someone who acts this way is risking eating forbidden foods, whether he convinces himself to believe or not. And in the end that is what matters, not necessarily the formal rules. Therefore the Shulchan Arukh ruled as it did, and the Rema was even more stringent.
(And by the way, this ruling in the Shulchan Arukh and in later authorities was accepted by all Jewish communities, was it not? If so, why should it be any less authoritative than a ruling of the Talmud?)