Yes, in practical terms. Aluminum is 100% recyclable, and melting it down does not change the metal itself, so it can be recycled again and again without losing its basic properties. A small amount is lost to oxidation during each melt, but the remaining metal is fundamentally unchanged.
Beverage cans are the clearest example: they are commonly recycled in a closed loop, going from used can back to new can sheet. According to industry data cited by the Aluminum Association, around 75% of all aluminum ever produced is still in use today.