Assalamu alaikum wa Rahmatullah
Is it permissible to publicise conversations or correspondence between two individuals by one of the participants without the explicit permission of the other? What if the matter is of major importance to islamic belief etc? Eg. if I were to have an email discussion with another Muslim without informing him that the discussion would be later publicised, can I later publish the full discussion on the public domain especially if the discussion exposes the false belief's of the individual?
Jazakkallahumal Khair.

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

It is permissible for a person to publish a conversation or correspondence between him and another person as long as there is no Sharee‘ah impediment such as being asked not to disclose it. In such a case, it is impermissible to publish it except with his permission, as the Prophet, sallallaahu ‘alayhi wa sallam, said, “There is no faith for a person who is not trustworthy, nor is there a religion for a person who breaks his covenants.” [Ahmad with a good chain of narrators]

This ruling applies only to the cases where there is no Sharee‘ah-approved benefit upon publishing such a conversation such as exposing the false beliefs or immorality of someone to warn Muslims against his evils.

In such a case, publishing these conversations may be obligatory as advice for the Muslims. The Prophet, sallallaahu ‘alayhi wa sallam, said, "Religion is advice". They said, "For whom?'' He replied, "For Allaah, His Book, His Messenger and for the leaders of the Muslims and their masses." [Muslim]

For more benefit, refer to Fatwa ‏92655.‏

Allaah Knows best.