# Quantum sheaves

Are the following definitions known?

Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:

(a) {0} and H lie in Sigma

(b) If {V_alpha} is a subset of Sigma then the closure of the sum of the V_alpha lies in Sigma

(c) If V, W lie in Sigma then their intersection lies in Sigma

Given a quantum topology, we introduce the notion of s a "quantum sheaf of operators"

s corresponds to every V in Sigma s(V), a C*-algebra of bounded operators on V. We demand the following conditions on s:

(i) If W lies in V, A lies in s(V) and leaves W invariant, then the restriction of A to W lies in s(W)

(ii) Suppose {W_alpha} is a subset of Sigma and V is the closure of their sum. Suppose A is a bounded operator on V which leaves every W_alpha invariant. Suppose further that the restriction of A to W_alpha lies in s(W_alpha) for every alpha. Then A lies in s(V)

(iii) Suppose {W_alpha} is a subset of Sigma and V is the closure of their sum. Consider C the subalgerba of s(V) consisting of operators of the form defined by condition (ii). Then the commutant of C lies within C.

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Are you assuming that all the $s(V)$ are realized on the same Hilbert space? (If not, then it would seem more natural to talk of $s$ as being a functor from the poset $\Sigma$ to the category of $C*$-algebras and $*$-homomorphisms) – Yemon Choi Nov 26 '11 at 22:16
In arxiv.org/abs/0709.4364 Heunen, Landsman and Splitters construct a sheaf over a C*-algebras for quantum logic. But I don't know whether their construction differs from your definition. – alexod Nov 26 '11 at 22:17
Instead of closed subspaces of a Hilbert space, you can phrase this in terms of closed ideals of the Von Neumann algebra of bounded operators on $H$. If you add the condition that a quantum topology must be closed under (pointwise) composition of operators, then it looks like you're asking for a sheaf on a quantale, for which there are various proposals; see e.g. [Borceux & Van den Bossche, Order 3:61--87, 1986]. – Chris Heunen Nov 28 '11 at 11:13