2 Second answer per comment

I have never seen finite-dimensional projective spaces defined by axioms, only by constructions of some kind from something else related to axioms. For example in algebraic geometry, you can define a projective variety as the Proj of a graded algebra, which could be adequately axiomatic. However, projective space in that setting comes from the relatively unsatisfying "axiom" that the algebra is freely generated from degree 1.

I was about to say that it would be very difficult to make axioms for a Hilbert-projective space that are very different from the axioms for a Hilbert space. But then I thought of a way to do it. A von Neumann algebra is a Banach *-algebra over $\mathbb{C}$ that satisfies the $C^*$ axiom and also has a predual as a Banach space. If $\mathcal{M}$ is a von Neumann algebra, it has a space $\mathcal{M}^\diamondsuit$ of pure normal states, by definition the extremal normal, normalized, positive dual vectors. This is a generalized projective space. In particular if $\mathcal{M}$ is a Type I factor — the conditions for which need no direct mention of Hilbert spaces — then von Neumann's theorem identifies $\mathcal{M}^\diamondsuit$ with the space of lines in a Hilbert space $\mathcal{H}$. (And of course $\mathcal{M}$ itself with $B(\mathcal{H})$.) No global phases are ever chosen in the definition.

Does this meet your requirements? My motivation is the fact that quantum probability is the correct probability theory for quantum mechanics, and that in quantum mechanics global phases are always irrelevant. Reflecting that, the global phase doesn't exist in the von Neumann definition of a state.

Qiaochu in the comments makes the point that there is another very important set of axioms for a projective space, namely the classical incidence axioms for a projective geometry. My favorite version is that a projective geometry is a spherical type $A_n$ building. These axioms are different in that they don't even pick a field beforehand. Indeed, there are projective planes that are not the standard projective plane over a field.

It would be interesting to make axioms for a topological type $A_\infty$ building corresponding to the projective space of a Hilbert space. It seems plausible, and it could be a very different model from the von Neumann algebra model. But maybe von Neumann, the person, is still there in this idea, because the incidence geometry of a Hilbert space is also known as quantum logic.

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I have never seen finite-dimensional projective spaces defined by axioms, only by constructions of some kind from something else related to axioms. For example in algebraic geometry, you can define a projective variety as the Proj of a graded algebra, which could be adequately axiomatic. However, projective space in that setting comes from the relatively unsatisfying "axiom" that the algebra is freely generated from degree 1.

I was about to say that it would be very difficult to make axioms for a Hilbert-projective space that are very different from the axioms for a Hilbert space. But then I thought of a way to do it. A von Neumann algebra is a Banach *-algebra over $\mathbb{C}$ that satisfies the $C^*$ axiom and also has a predual as a Banach space. If $\mathcal{M}$ is a von Neumann algebra, it has a space $\mathcal{M}^\diamondsuit$ of pure normal states, by definition the extremal normal, normalized, positive dual vectors. This is a generalized projective space. In particular if $\mathcal{M}$ is a Type I factor — the conditions for which need no direct mention of Hilbert spaces — then von Neumann's theorem identifies $\mathcal{M}^\diamondsuit$ with the space of lines in a Hilbert space $\mathcal{H}$. (And of course $\mathcal{M}$ itself with $B(\mathcal{H})$.) No global phases are ever chosen in the definition.

Does this meet your requirements? My motivation is the fact that quantum probability is the correct probability theory for quantum mechanics, and that in quantum mechanics global phases are always irrelevant. Reflecting that, the global phase doesn't exist in the von Neumann definition of a state.