Extension theorems (vector measures on quantum logics)
Anna Avallone, Jan Hamhalter · Czechoslovak Mathematical Journal · 1996
INTRODUCTION AND PRELIMINARIESVector valued measures and their extensions have been widely studied by many authors (see e.g.[1,2,4,5,7,8, 10,11,12,16,19,20]).The aim of this paper is to complete and generalize known results concerning extensions of various types of vector and group-valued measures defined on Boolean algebras to larger orthomodular structures.Our discussion falls into three parts.The first part is devoted to extensions of orthogonal measures.(Measures of this type play important role in the noncommutative probability theory and foundations of quantum physics-see e.g.[4, 14] for systematic treatment.)It has been proved in [12] that, if H is a finite dimensional Hilbert space and L is a logic, then the condition that L is H-rich (i.e.L has enough H-valued orthogonal measures) is equivalent to the following extension property: for any Boolean subalgebra B of L and any orthogonal measure m: B --> H, there exists an orthogonal measure m: L -> H extending m.We show that, if H is infinite dimensional, then the condition of H-richness is not sufficient for the above extension property In this connection we prove general extension theorem for orthogonal measures having values in arbitrary Hilbert space.As a corollary we show that every orthogonal measure on a Boolean subalgebra of the projection logic P(M) of a von Neumann algebra M extends to an orthogonal measure on P(M) with values in some (generally larger) Hilbert space.In the second part of this note we study extensions of vector measures defined on the centre of a given orthomodular lattice L. In this special case we can get extension result without assuming that L has a large set of measures.In particular we proveThe second author