Multiplication in Riesz spaces

Abstract

A.G. Vulih has shown how an essentially unique intrinsic multiplication can be defined in certain types of Riesz spaces (vector lattices) L. In general, the multiplication is not universally defined in L, but L can always be imbedded in a large space L# in which multiplication is universally defined. If ф is a normal integral in L, then ф can be extended to a normal integral on a large space L1(ф) in L#, and L1(ф) may be regarded as an abstract integral space. A very general form of the Radon-Nikodym theorem can be proved in L1(ф), and this can be used to give a relatively simple proof of a theorem of Segal giving a necessary and sufficient condition that the Radon-Nikodym theorem hold in a measure space. In another application, the multiplication is used to give a representation of certain Riesz spaces as rings of operators on a Hilbert space.