I needed to write a business e-mail to my US partner just after Christmas day. Are there any established forms of such a greeting?
I hope you had a nice Christmas
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"I hope you had a nice Christmas" or "I trust you had a nice Christmas" would both suffice. The latter might sound a bit overly-formal to some ears, but if it's a business contact you don't know outside of business that's not necessarily a bad thing.
Much is made in some quarters about whether it is better to refer to Christmas or the holidays generally, with differing opinions on both sides.
Personally as a non-Christian who does celebrate a different holiday near the same time of the year, I find nothing objectionable about people wishing me a happy Christmas - being happy at Christmas is definitely preferable to the alternative, after all.
Still, if you know that they celebrate Christmas or if they have mentioned Christmas previously as the reason they will be un-contactable for a while, then go with Christmas. If you know they celebrate another holiday (solstice and Chanukkah were both celebrated recently) then do mention the holiday they celebrated. If you're unsure then something like "I trust you enjoyed the break and are keen to start on the new year's projects" can avoid mentioning a particular holiday without sounding like you're avoiding it (there is a minority who consider anybody enjoying a holiday near Christmas that isn't Christmas to be a "war on Christmas" and get upset about avoiding being overly specific).
All that said though, "I hope you had a nice Christmas" can't go far wrong.
Also, if English is your second language and they know where you are from, a native greeting in your own language can be a nice touch, especially if context or similarity in etymology makes it easy to guess the meaning of.