Spans of quantum-inequality projections

Abstract

A hereditarily atomic von Neumann algebra $A$ is a $W^{\ast }$ 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${}^{\ast }$-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${}^{\ast }$ topology.

18 pages + references