From Acts of the Apostles, by William S. Kurz, SJ, commenting on Acts 3:13-15, 17-19
With the phrase the God of our ancestors, Peter emphasizes his shared Jewish identity with his listeners. The expression underscores the continuity between God’s present action through the apostles and his former deeds for his people in the Old Testament. The God who is acting through the apostles is the God of Abraham, [the God] of Isaac, and [the God] of Jacob (see Luke 13:28; 20:37; Acts 7:32), who revealed himself to Moses at the burning bush (Exod 3:15). The same God who made promises to the patriarchs of Israel—the only God—continues to act through Jesus his Son.
This healing is a sign that God has glorified his servant Jesus, whom you, his Jewish compatriots, had rejected before Pilate (see John 1:11). Though “servant” can also be translated “child,” here it alludes to Isaiah’s prophecies of the Suffering Servant, who would suffer for the sins of the people but be glorified by God (Isa 52:13–53:12; Acts 8:32–35).
Peter calls Jesus the Righteous One, a title used by the centurion at the cross who recognized Jesus’ innocence (Luke 23:47). In sharp contrast, the listeners had a murderer, Barabbas, released (Luke 23:18–19) and killed the author of life. The word for “author” can mean that Jesus is the cause or originator of life (see John 1:1–4) or that he is the pioneer or leader of life, inasmuch as he is the first to rise to life after death (see Col 1:18). Although the Jews who clamored for Jesus’ crucifixion chose death over life, God reversed their decision and raised him from the dead, of which the apostles are witnesses.