If a person comes to the synagogue on Friday night after the congregation has already accepted Shabbat through the singing of Lecha Dodi, he may not recite the Minha prayer in the synagogue. Once the congregation has accepted Shabbat, it is forbidden to recite a weekday prayer in the synagogue while they pray a Shabbat service. The individual must therefore recite Minha either in the hallway or in a separate room in the building, rather than in the sanctuary where the congregation recites the Shabbat prayer.
Furthermore, if a person comes into the synagogue on Friday night as the Hazzan recites "Barechu" at the beginning of the Arbit service, and he joins the congregation in answering, "Baruch Hashem Ha’meborach," then he may no longer recite Minha. The response of "Baruch Hashem Ha’meborach" constitutes an acceptance of Shabbat, and one therefore cannot recite thereafter the Friday afternoon service. According to Hacham Ovadia Yosef (Yehave Da’at 6:18), this applies even if one had specific intention not to accept Shabbat. Hacham Ovadia ruled that the response of "Baruch Hashem Ha’meborach" constitutes an acceptance of Shabbat regardless of the person’s intent, and thus even if he had in mind not to accept Shabbat he may not recite Minha. Therefore, one who hears the Hazzan reciting "Barechu" on Friday night before he recited Minha should not respond. If he did respond to "Barechu," such that he can no longer recite Minha, then he recites an extra Amida at Arbit to make up for the missed Minha prayer.
The Halachic authorities address the question of whether one may begin reciting Minha in the synagogue if the congregation has yet to recite Lecha Dodi, but will reach Lecha Dodi before he completes the Amida. On the one hand, since the congregation has yet to accept Shabbat, we might say that he is allowed to begin the weekday Minha prayer. On the other hand, one might contend that he should not begin Minha, since he will likely be reciting part of the Minha service after the congregation accepts Shabbat.
The Bet Yosef (work by Maran, author of the Shulhan Aruch) cites two conflicting rulings of the Terumat Ha’deshen (Rabbi Yisrael Isserlen, Germany-Austria, 1390-1460) in this regard. In his published work of responsa, the Terumat Ha’deshen rules that one may, in fact, begin reciting Minha even if the congregation will likely accept Shabbat before he completes the Amida. In one of his letters, however, the Terumat Ha’deshen indicates that to the contrary, one may not begin Minha in the synagogue in this case unless he figures that he can complete the Amida before the congregation accepts Shabbat.
After citing these conflicting rulings, the Bet Yosef establishes that whenever an author issues contradictory rulings in a personal letter and in his published work, the ruling in his published work is deemed more authoritative. We may assume that before publishing an author carefully reviews the manuscript several times in order to ensure accuracy, whereas in personal correspondence less caution is taken. In this case, then, Halacha follows the ruling that appears in the published responsa of the Terumat Ha’deshen, which allows one to begin reciting the Friday Minha service in the synagogue even if the congregation will likely accept Shabbat before he completes the Amida. This is, indeed, the ruling of Hacham Ovadia Yosef in his work Hazon Ovadia (208:90).
(See also "Reciting Minha When the Congregation Recites Arbit," dated July 2, 2008.)
Summary: One who arrives late in the synagogue on Friday night may not recite Minha in the synagogue if the congregation has already accepted Shabbat through the recitation of Lecha Dodi; he must recite Minha either in the hallway or in another room. If the congregation has not yet accepted Shabbat, then he may begin reciting Minha in the synagogue, even if he suspects that the congregation will accept Shabbat before he completes the Amida. If one who has yet to recite Minha on Friday afternoon hears the Hazzan reciting "Barechu," he should not respond with the congregation. If he does respond, then he may no longer recite Minha, and should instead recite an extra Amida at Arbit to make up for the missed Minha.
