Today a paper was posted on the arXiv by Blackadar, Farah and Karagila (arXiv:2304.09602) that examines Hilbert spaces and C*-algebras in ZF set theory without the assumption of any Axiom of Choice (not even with Countable Choice).
Since many standard tools from functional analysis like the Hahn-Banach theorem or the Baire category theorem fail without Choice, one has to be very careful in proving even the basic facts; and this is on top of the fact that, for example, fundamental notions like infinite have different definitions without Choice.
Some exotic phenomena that can happen now, are the following:
- There exist sets \(X\) (with strange set-theoretic properties without Choice) such that the Hilbert space \(H = \ell^2(X)\) is infinite-dimensional but every closed subset of it has either finite dimension or finite codimension. Further, the C*-algebra of all bounded, linear operator \(B(H)\) is in this case equal to \(K(H) + \mathbb{C} \cdot \mathrm{id}\).
- There exist sets \(X\) (with strange set-theoretic properties without Choice) such that the spectrum of each bounded linear operator on \(H = \ell^2(X)\) consists of only finitely many points which are all eigenvalues.
- In certain models of ZF the commutative C*-algebra \(\ell^\infty(\mathbb{N}) / c_0(\mathbb{N})\)
- … has no non-trivial bounded linear functionals,
- … has no non-trivial representation on a Hilbert space,
- … is not isomorphic to \(C(X)\) for a compact Hausdorff space \(X\).