Spans of quantum-inequality projections
Alexandru Chirvasitu
TL;DR
The paper studies projections in $A \overline{\otimes} A^{\circ}$ for hereditarily atomic von Neumann algebras $A$, interpreting them as quantum binary relations and focusing on the quantum-inequality projections $p\otimes(1-p)$. It then analyzes the symmetrized linear span of these projections under the unitary symmetry of $A$, proving that the symmetrized span $\textsc{INQ}(A)_+$ equals the symmetric part of the joint kernel of multiplication and opposite multiplication, i.e. $(\ker\mu \cap \ker\mu^{\circ})_+$, with the result derived via representation-theoretic decompositions under the compact group $\mathrm{U}(A)$. The authors reduce to the finite-dimensional matrix setting to obtain a precise identification in the symmetric sector, and note that the analogous antisymmetric statement does not hold in general, though it does for matrix algebras $A \cong M_n$. Beyond the main span result, the paper investigates the joint-continuity of unitary conjugation actions on tensor powers, introducing the notion of virtually classical quantum sets and providing a classification: joint continuity holds precisely in the finite-dimensional-by-abelian scenario, while it can fail in infinite-dimensional cases. These results illuminate the interplay between linear (quantum superposition) and multiplicative (observables) structures in quantum-set predicate calculus via a robust symmetry-based framework.
Abstract
A hereditarily atomic von Neumann algebra $A$ is a $W^*$ product of matrix algebras, regarded as the underlying function algebra of a quantum set. Projections in $A\overline{\otimes}A^{\circ}$ are interpreted as quantum binary relations on $A$, with the supremum of all $p\otimes (1-p)$ representing quantum inequality. We prove that the symmetrized weak$^*$-closed linear span of all such quantum-inequality projections is precisely the symmetric summand of the joint kernel of multiplication and opposite multiplication, a result valid without the symmetrization qualification for plain matrix algebras. The proof exploits the symmetries of the spaces involved under the compact unitary group of $A$, and related results include a classification of those von Neumann algebras (hereditarily atomic or not) for which the unitary group operates jointly continuously with respect to the weak$^*$ topology.