Shared ground
Peter places his Jerusalem hearers inside Israel’s Scripture story. He calls them “sons” of the prophets and of the covenant God made with “our fathers,” then anchors that claim by citing God’s promise to Abraham: blessing is meant to reach “all the families of the earth.” (Explicit textual claim.)
Peter then connects that covenant promise to God’s action in Jesus. God “raised up his servant, Jesus,” and “sent him to you first.” The purpose of this sending is “to bless you,” and Peter defines that blessing as God “turning away everyone of you from your wickedness.” (Explicit textual claim.)
Where interpretation differs (only where needed)
Who “you” includes. Many think Peter is addressing a broadly Jewish crowd in Jerusalem associated with the temple and Israel’s covenant story. Others argue the wording could be aimed more narrowly at those most responsible for rejecting Jesus earlier in the speech, even if others are listening.
What “raised up” emphasizes. Some take it mainly as “raised from the dead,” highlighting resurrection. Others hear it as “brought onto the scene / appointed,” stressing God commissioning Jesus as the servant; in this view resurrection is still true in Acts, but not necessarily the only idea here.
How the “turning away” works. Some read v. 26 as describing God’s active work in changing people. Others read it as describing what God’s blessing produces as people respond and turn; the line can be read as closely linking God’s action and human response without spelling out mechanics.
Why the disagreement exists
The key phrases are compact and can carry more than one normal sense in biblical Greek and in Scripture-shaped speech (“sons of…,” “raised up,” “sent…first,” “turning away”). Also, Peter is summarizing big covenant themes quickly, so the text leaves some details unstated.
What this passage clearly contributes
It ties the message about Jesus to the Abraham covenant and to the prophets, not as a new religion detached from Israel’s story. It also frames the initial audience as Israel first (“sent…to you first”) while insisting the Abraham promise was always aimed at “all families of the earth.” Finally, it defines “blessing” in moral-spiritual terms: God’s intended good includes people being turned from concrete wrongdoing, person by person.