What Information Can Be Recovered from the Abelian Subalgebras of a von Neumann Algebra

Abstract

As explained in [26], the motivation for constructing topos quantum theory is to render quantum theory more “realist”. This is done by expressing quantum theory in terms of similar mathematical constructs used to formalise classical physics, in the hope that the ensuing interpretation would be more “realist” (as is the case for classical theory). This is achieved by describing quantum objects in terms of presheaves in the topos $\mathbf { Sets}^{\mathcal {V}(\mathcal {H})}$, where $\mathcal {V}(\mathcal {H})$ is the poset of abelian von Neumann subalgebras of the algebra of bounded operators $\mathcal {B}(\mathcal {H})$. Each such abelian subalgebra $V\in \mathcal {V}(\mathcal {H})$ represents a classical snapshot since it contain only simultaneously measurable observables. Hence a quantum object can be seen as a collection of classical approximation “glued” together by the categorical structure of the base category $\mathcal {V}(\mathcal {H})$. The quantum information is then contained in the categorical structure of $\mathcal {V}(\mathcal {H})$ which is also reflected at the level of presheaves. The question that then comes to mind is the following: given a general von Neumann algebra $\mathcal {N}$ and its collection of abelian subalgebras $\mathcal {V}(\mathcal {N})$, how much of $\mathcal {N}$, if any, can be reconstructed from $\mathcal {V}(\mathcal {N})$? This question was asked in [37]. There it was shown that, if the initial von Neumann algebra $\mathcal {N}$ is abelian, then it can be completely reconstructed from the poset of its abelian von Neumann subalgebras. However, if the algebra $\mathcal {N}$ is not abelian, then it can only be reconstructed up to its Jordan structure. This is because both $\mathcal {N}$ and its opposite ${\mathrm {op}}(\mathcal {N})$ have the same collection of subalgebras but they are not necessarily isomorphic to each other [9].