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In P2P streaming networks, the main sources of illegal media contents sharing are streaming clients who ignore copyright laws and provide contents deliver services. To stop illegally media contents sharing activities with the boundary of a P2P streaming network, we propose a time-space dynamic trust model for legalizing peers' content delivery services. We incorporate time dimension using time-frame, which captures experience and recommendation's time-sensitivity. At the same time, we introduce space dimension using IP addresses, which reflects the peers' physical locations and relations. Together, these two dimensions are adjusted using positive feedback control mechanism, thus, trust valuation can reflect the dynamics of the trust environment. Theoretical analysis and simulation results show that, out proposed trust model has advantages in modeling time-space dynamic trust relationship. It is capable to detect and penalize the illegally media contents sharing peers, as well as those that exhibit malicious behavior. Moreover, the trust model can filter out dishonest peers effectively.