Upvote:2
You will be stamped into the Schengen Area in Amsterdam. Except if you hit a sporadic spot check, that's your last encounter with passport control, whether you stay in the airport or go into town.
Upvote:9
Amsterdam is in the Schengen area, and as a US citizen you can enter the Schengen area for short visits (of up to 90 days of any 180) without having a visa.
This is the same rule that allows you to go to Norway without a visa, as Norway is also in the Schengen area.
(In fact, no matter whether you leave the airport or not, you will pass through the common Schengen immigration point in Amsterdam, and will not pass through immigration when you arrive in Norway).