Can I cause a Melacha to happen in a place where Shabbat has already started? For example, if I live in New York, could I send a fax (causing the machine to print a document) to Israel on Friday afternoon, before it is Shabbat for me, but after Shabbat has started there?
The Shut Hor Yizchak (Hor Hachaim 157) says that not only it is OK, but the question not even starts ("he did nothing"). The example he gives is to call from Israel to the States on Motzei Shabat. In Israel it is after Shabat and in the States the Shabat is still on. He adds that it is even permissible to phone a non Jew and ask him to do work for you.
I read a shiur that discusses whether the commandment to rest on Shabat is for the Jew or for Creation (World?). For those who are of the opinion that the commandment is for Creation to rest, this will not be permissible. The Grm"o Fraind is brought as a posek that follows this opinion, but the majority of the poskim permit it.
Rav Hershel Schacter discusses a Kasav yad of the Rambam which could assur such ideas. http://www.ou.org/torah/article/let_my_people_know inyun starts at 8:50 where he discusses this peirush HaRambam and Rav Moshe on setting up something that causes a melacha on Shabbas and he comes out that l'chumra it should not be done and the OU tries to avoid sending emails where it is shabbas somewhere else.
If you send a fax on Friday (before sunset in your timezone) to Israel, or anywhere else in the world, such that it is after the Sabbath has commenced in that time zone, then you are causing the same amount of work to be done as setting electronic timers to turn off and on the lights during the Sabbath. So like @rony says, if you find that degree of activity to be acceptable, the electromagnetic activity generated by transmission of a fax, and the electro-mechanical activity of the fax machine applying heat or ink to paper and producing the fax, then it is acceptable. It means that you are of the opinion that the Sabbath is not for all Creation to rest.
As another said, the fax will certainly not be read by an observant Jew who receives that fax during the 24 hour interval of the Sabbath. It will just sit on the fax machine untouched. Once the recipient is able to read the fax, he or she will not contact you until after the Sabbath wherever you are. So there seems little point in sending the fax until after the Sabbath in your time zone, if it is regarding a matter for which you want to transact business with the recipient.
The reverse scenario would be equally applicable. You are in some part of the world after the Sabbath has ended. You then telephone, or fax, or send an email, to someone in a timezone where it is still Sabbath. The action is not an issue, a wrong action in its own right. But there should be no expectation of any response, neither answering the phone, nor reading of the fax or email and acting on it, until the Sabbath has ended for the recipient in their time zone, however many hours it is behind your time.
Final scenario: Let us assume that the recipient needs the fax to transact business with a third party. In that case, sending the fax so that either the recipient can act on it after sunset on Saturday, or a non-Jewish employee of the recipient can act on it during the Sabbath (and transact business with other non-Jews, on the Sabbath) is acceptable.
Shmiras Shabbos Kehilchasa chapter 31 siff 28 quotes Rabbi S.Z. Aurbach as saying this is allowed.
He also quotes him as saying the Jew who recieves the fax may not read it.
I asked Rabbi Dovid Feinstein about this once and he said one may not send a fax from America to Israel once it is shabbos for them.