Rabbinic sources place some responsibility for the destruction of the Temple on the shoulders of the Israelites themselves: In the Babylonian Talmud (Yoma 9b) we learn that the Second Temple was destroyed because of baseless hatred among the different Jewish communities living in the Land of Israel at the time.
Suddenly a fast that sounds archaic and anachronistic (why should we still mourn the loss of a building in which animal sacrifice was practiced?) becomes urgently contemporary: sadly, it is a phenomenon we experience all over the Jewish world, not just in Israel. One group, or denomination of Judaism condemns, rejects, belittles others because their practice or their theology, or their norms of dress and behavior, or their choice of rabbis, does not meet that groups standards. We Jews are not terribly successful at loving one another . . . and sometimes it's not even a matter of cross-denominational disdain . . . sometimes it's just about "the shul I wouldn't set foot in."
So should we mourn on the 9th of Av?
I'd say: a little bit. In a teshuvah written almost 30 years ago, Rabbi Tuvia Friedman, z"l* writing for the Israeli Masorti /Conservative Law Committee (Va'ad Halakahah): There is a clear historical precedent of canceling a fast on days in which the Jewish people was saved from a disaster. We have been so fortunate as to witness the founding of the State of Israel, where Jews are sitting on their land as a sovereign people. In light of this decisive change in the history of the Jewish people, I propose marking this change by not completing the fast of Tish'a b'Av, and concluding the fast with a Minhah Gedola service.
But I would also say that our observance of the fast of the 9th of Av should not end with the shortened fast, but should serve as a day to commit to pursuing the value of "k'vod hadadi", of mutual respect among the various denominations that comprise the Jewish world.
There may be room, therefore, for some mourning on this historic day of remembrance, but let us use the message of this day to develop what unites us as Jews and to approach each other with mutual respect and a sense of being one very extended family.