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I want to know if I can send a private message to a fan of my page as the sender is the page itself which is created by me so that the fan can not identify me?
As of now you can not initiate a conversation with a fan. If the person starts the conversation, yes, you can. But you can't send the first message.
What you can do is create a post (on your page) for that person asking him/her to contact you and tag the person.
Facebook has recently changed this process and you can now send messages as a page admin.
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