Spans of quantum-inequality projections

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 ⊗ ̄ A ° are interpreted as quantum binary relations on A, with the supremum of all p ⊗ (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.