Stott's reading this morning (p. 293 in Through the Bible Through the Year) notes that Jesus' brothers are among those who, with the remaining apostles,"the women[,] and Mary the mother of Jesus," had "all joined together constantly in prayer" during the period between the Ascension and the coming of the Holy Spirit. (Acts 1:14) I had never understood what happened to turn the brothers around, as the last time we see them before this point is during Jesus' pre-Calvary ministry. They were then apparently in opposition to him (Luke 8: 19-21) and did not believe in him (John 7:5).
Stott notes that "they had finally come to faith because of the private resurrection appearance that Jesus had granted to James the Lord's brother (1 Cor. 15:7)." Here is that reference in context:
5:1 Now I want to make clear for you, brothers and sisters, the gospel that I preached to you, that you received and on which you stand, 15:2 and by which you are being saved, if you hold firmly to the message I preached to you – unless you believed in vain. 15:3 For I passed on to you as of first importance what I also received – that Christ died for our sins according to the scriptures, 15:4 and that he was buried, and that he was raised on the third day according to the scriptures, 15:5 and that he appeared to Cephas, then to the twelve. 15:6 Then he appeared to more than five hundred of the brothers and sisters at one time, most of whom are still alive, though some have fallen asleep. 15:7 Then he appeared to James, then to all the apostles. 15:8 Last of all, as though to one born at the wrong time, he appeared to me also. [NET Bible; my emphasis]
The following note to 1 Cor. 15:7 is in my NIV Study Bible, as to James:
"Since this James is listed in addition to the apostles, he is not James son of Zebedee or James son of Alphaeus (Mt. 10:2-3). This is James, the half-brother of Jesus (Mt 13:55), who did not believe in Christ before the resurrection (Jn 7:5) but afterward joined the apostolic band (Ac 1:14) and later became prominent in the Jerusalem church (Ac 15:13). It is not clear in Scripture when and where this appearance to James occurred. "