proving $IP^\star = NP$
If $x \in L$ then the probability that $(P,V)(x,r) = 1$ is positive, where $r$ is the randomness involved; the probability is over the choice of $r$. In particular, there is some $r$ such that $(P,V)(x,r) = 1$.

In contrast, when $x \notin L$, the probability that $(P,V)(x,r) = 0$. That is, there is no $r$ such that $(P,V)(x,r) = 1$.

States differently, when $x \in L$, there is some conversation for which the verifier outputs 1, whereas when $x \notin L$, there is no such conversation. Since the verifier runs in polytime, the conversation has polynomial length, and can be verified to be a valid conversation in polynomial time. This puts $L$ in NP.



What is the difference between IP* and IP? IP* has perfect soundness – when $x \notin L$, the probability of error is 0 – whereas IP has neither perfect soundness nor perfect completeness. This makes IP much stronger.