Infinite tensor products of commutative subspace lattices

Abstract

Every infinite tensor product of commutative subspace lattices is unitarily equivalent to a certain lattice of projections on L 2 ( X , ν ) {L^2}(X,\nu ) , where ( X , ν ) (X,\nu ) is an infinite product measure space. This representation reflects the structure of the individual component lattices in that the components of the tensor product correspond to the coordinates of the product space. This result generalizes the similar representation for finite tensor products. It is then used to show that an infinite tensor product of purely atomic commutative subspace lattices must be either purely atomic or noncompact, and in the latter case the algebra of operators under which the lattice is invariant has no compact operators.