Is there a way that Appian can connect to a different mailbox, and send mails via that connected mailbox?
This mailbox is application specific, that means only mails which are sent by certain processes should be using this. No other Appian processes should use this mailbox.
Also, if such kind of setup is successfully established, when someone replies back, will the mail hit the same mailbox?
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You can use Appian's custom sender option available in the send email smart services to configure your mailbox's address.
But in order to prevent spoofing Appian explicitly sets the sender and return path to its own email server (if on cloud) this will prevent any replies from reaching your mailbox.
source:
To avoid it you can setup a custom mail server to allow Appian to send emails from your mail domain.
please refer this documentation, but it's a lengthy process.