Quantum field theory in curved spacetimes is often described in the algebraic approach, which consists of describing observables as elements of a certain $*$-algebra. To recover the notion of a Hilbert space, one represents this algebra as operators acting on said Hilbert space. Given a state on the algebra, the GNS construction allows one to obtain a particular representation of this algebra. Given two different states, such as the Minkowski and Rindler vacua in Minkowski spacetime, it might happen that the representations are not unitarily equivalent.
What I find curious, though, is the existence of the following theorem in Functional Analysis (see Kreyszig's Introductory functional analysis with applications Theorem 3.6-5)
Two Hilbert spaces $H$ and $\tilde{H}$, both real or both complex, are isomorphic if and only if they have the same Hilbert dimension.
The Hilbert dimension is the cardinality of an orthonormal basis of the Hilbert space.
Now my question is: how to make sense of this? For example, both the Minkowski and Rindler vacua lead to representations in Fock spaces. Isn't a Fock space always separable, and hence has Hilbert dimension $\aleph_0$? Shouldn't then any two Fock space representations be unitarily equivalent? In particular, shouldn't the Minkowski and Rindler vacua lead to unitarily equivalent representations? Why don't they?