The Pri Megadim (Eshel Avraham 53:19) rules that a convert is obligated to ensure that his or her fellow-Jews observe the mitzvot just as it is incumbent upon a born Jew to do. This is called Arvut (shared responsibility). He infers this from the fact that a convert can lead the prayers which includes saying the repetition of the Amidah. (See the Mishnah quoted above as well as O.C. ibid.) The repetition of the Amidah was instituted so that people who did not know how to pray on their own could listen to the repetition and fulfill their obligation. The idea that one who has already recited a certain prayer can repeat it on behalf of someone else is based on the fact that all Jews are Areivim (responsible) for one another. Therefore, if a convert can repeat the Amidah for others even though he already prayed, it must be that he is considered responsible for other Jews as well.
This is also the opinion of the Maharit (Rabbi Yosef Trani 1568 - 1639) on Kiddushin 70b.
Some say that converts do not have Arvut (Tosfos D.H. Kashim Gerim on Kiddushim ibid and Rashi D.H. Kesapachat on Niddah 13b).
Rabbi Shlomo Drimmer (1800 - 1872 of Skolia, Ukraine) writes (Bais Shlomo, O.C. 14) that even according to the opinions that converts do not have shared responsibility, a convert may be a chazzan.
