The space of measurement outcomes as a spectrum for non-commutative algebras

Bohrification defines a locale of hidden variables internal in a topos. We find that externally this is the space of partial measurement outcomes. By considering the double negation sheafification, we obtain the space of measurement outcomes which coincides with the spectrum for commutative C*-algebras.

💡 Research Summary

The paper develops a novel spectral construction for non‑commutative C*‑algebras by employing the technique of Bohrification within a topos‑theoretic framework. Starting from a C*‑algebra A, the authors consider the poset 𝒞(A) of its commutative unital sub‑algebras, ordered by inclusion. By viewing 𝒞(A) as a site, they form the presheaf topos 𝒯 =