Building cooperative communities has always been a crucial problem for human societies. Much research suggests that cooperation is facilitated by knowing who exactly is a cooperator and defector, and being able to respond accordingly. As such, anonymous games is thought to hinder cooperation. Here, we show that this conclusion is altered dramatically in the presence of conditional cooperation norms and heterogeneous beliefs about others' behaviours. Specifically, we show that inaccurate beliefs about other players' behaviours can in fact help foster and stabilize cooperation via social norms. To show this, we combine the population dynamics of a social community with the game theory of interactions within the community. In our model, individuals can join a community based on beliefs generated by public signals regarding the level of cooperation within, and decide to cooperate or not depending on these beliefs. These signals may overstate how much cooperation there really is. We show that even if individuals eventually learn the true level of cooperation, the initially false beliefs can trigger a dynamic that sustains high level of cooperation within the community. We also characterize how the rates of joining, leaving, and learning in the community affect the cooperation level and community size simultaneously. Our results illustrate how false beliefs in the presence of conditional cooperation norms can help build up cooperative communities.