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It is important to enable peers to represent and update their trust in other peers in open networks for sharing files, and especially services. We propose a Bayesian network-based trust model and a method for building reputation based on recommendations in peer-to-peer networks. Since trust is multifaceted, peers need to develop differentiated trust in different aspects of other peers' capability. The peer's needs are different in different situations. Depending on the situation, a peer may need to consider its trust in a specific aspect of another peer's capability or in multiple aspects. Bayesian networks provide a flexible method to present differentiated trust and combine different aspects of trust. The evaluation of the model using a simulation shows that the system where peers communicate their experiences (recommendations) outperforms the system where peers do not share recommendations with each other and that a differentiated trust adds to the performance in terms of percentage of successful interactions.