Is it permissible to suffice with playing a recorded adhan when the time for prayer begins, if no one from the mosque's congregants performs the adhan?

The adhan (call to prayer) broadcast from a recording device is not sufficient to replace the real, lawful adhan, because it is not a true adhan, but rather a stored sound. The adhan is an act of worship that requires both action and intention. Transmitting the adhan via a recording is not permissible as a substitute for the lawful adhan, because the lawful adhan is a remembrance of and praise for Allah, and it must involve action. If we say that what is recorded is not a legitimate adhan, then responding to it is not prescribed. As for what is transmitted live, it is the voice of the mu'adhdhin (caller to prayer) who is performing the lawful adhan.