TITLE: Ax–Grothendieck and the Garden of Eden
QUESTION [27 upvotes]: It's an obvious consequence of the pigeonhole principle that any injective function over finite sets is bijective. But there are some similar results in different areas of mathematics that apply to less-finite settings.
In algebraic geometry, the Ax–Grothendieck theorem states (if I have it correctly) that any injection from an algebraic variety over an algebraically closed field to itself is bijective; the standard proof involves some sort of local-global principle together with the same fact over finite fields.
In the theory of cellular automata, the Garden of Eden theorem states that any injective cellular automaton (over an integer grid of some fixed finite dimension, say) is bijective; the standard proof involves again the same fact for finite sets of cells together with a limiting argument that shows that for large enough bounded regions of an unbounded grid, the boundary of the region has negligible effect compared to the interior.
Is there some way of viewing these three injective-bijective statements (or others) as instances of a single more general phenomenon?

REPLY [3 votes]: In the theory of von Neumann algebras, there is a similar phenomenon.
Let $M$ be a type $II_1$-factor. That is, it's an infinite dimensional von Neumann algebra with center $\mathbb C$, and with an (everywhere defined) trace $tr:M\to \mathbb C$.
Then there is a complete invariant of $M$-modules called the von Neumann dimension.
This invariant takes values in $\mathbb R_{\ge 0}\cup \{\infty\}$ and can take any value in that set. It has the property that any isometric map $H_1 \to H_2$ between modules of the same dimension is actually a unitary isomorphism (except if the von Neumann dimension is $\infty$, in which case, that's not true).
In particular, if $H$ is an $M$-module of finite von Neumann dimension, then any isometry (not assumed to be surjective) is actually a unitary isomorphism.