It might seem a bit of an odd one, but - Well, here's the two sides of the argument, as I see it:
On the one hand, the necklace surrounds whoever wears it with breathable air. Breatheable air isn't really a medium most adventurers can swim through, so it seems reasonable that an adventurer who puts on a necklace and then jumps out of their boat would find themselves surrounded entirely by air on all sides - and so immediately falling to the lake bottom, the water beneath them continually replaced with more empty air for them to fall through. (Presumably, they'd also emit a great plume of bubbles once underwater as air escaped from their continually-replenished personal air supply.)
On the other hand, the bubble is described as a "shell" of air; Maybe the intent is that a "shell" surrounds the adventurer like some kind of suit, and "sticks" to them instead of bubbling up? If that's the case, I feel like maybe the adventurer would float and able able to swim as normal - or rather, to "fly" inside their bubble - since force would transfer from them, to the bubble, and thus to the water that surrounds it, and vice versa. But that also seems weird, since you'd expect the shell of air to also repel other objects, including adventurer's own clothing and equipment and the ground, and that seems like too important a detail to leave out of the object description.
As a GM, I tend towards simulationist play styles where players are rewarded for coming up with creative uses for magic items - but to do that, I need to have some idea as to how those magic items work.
Is there any text that clarifies how a necklace of adaptation does its thing? Can an adventurer wearing one swim, or are they instead able to walk along the lake bottom? Either option sounds plausible and useful. Which, if any, is correct?
I'm tagging this with 3.5e and Pathfinder because those are the games I'm currently running, but I'm perfectly willing to adapt answers from other editions if there's nothing more immediately relevant. Or from published novels set in official campaign settings, for that matter.