Rav Shemuel Pinchasi, in his work on the laws of Shabbat, addresses a number of interesting cases involving telephone calls made from a place where it is not Shabbat to a place where it is Shabbat, and vice versa. We present here a number of important Halachot he discusses in this context. (These Halachot appear as well in the work "Melachim Omenayich.")
If a gentile in the United States makes a phone call on Shabbat afternoon to a Jew in Israel, after Shabbat had already ended in Israel, it is permissible for the Jew to receive the call and speak with the gentile. Since Shabbat had already ended for the Jew, and the call was made on Shabbat by a gentile, answering the call entails no prohibition. By the same token, a Jew in the United States may make a call on Friday afternoon to a gentile in Europe or somewhere else where Shabbat has already begun. Since the Jew makes the call before Shabbat has begun in his area, and he calls a gentile, and not a Jew, no prohibition is involved even though Shabbat has begun in the area to where the call is placed.
Likewise, Rav Pinchasi rules that one may send a fax from the United States on Friday afternoon to Israel, even though Shabbat has already begun in Israel. Conversely, a Jew in Israel may send a fax on Motza'ei Shabbat to the United States, even though Shabbat has yet to end in the U.S. In both these cases, the Jew sends the fax when it is not Shabbat, and thus he does not violate Shabbat even though the fax arrives in an area where Shabbat is observed. Nevertheless, Rav Pinchasi advises that one rely on this leniency only in situations of particular necessity.
Finally, a Jew in Israel who trades stocks in the New York Stock Exchange must instruct his broker not to purchase stocks on his behalf on Friday once Shabbat has begun in Israel. Even though Shabbat has yet to begin in New York, it is forbidden to allow stocks to be purchased on one's behalf during the period he observes as Shabbat.