Representations of positive projections 1 Introduction - Mathematics ...

of F given by 1 = fF Y : F 2 g, and similarly is identi ed with
the corresponding -subalgebra 1 of F. Assuming in addition that is a
- nite measure on (Y ) and that 0 m 2 M ( F) satis es
Z
Y
m (x y) d (y) =1 8x 2 X, (6)
we de ne the linear operator Rb : Mb ( F) ! Mb ( F) by
Rbf (x y) =
Z
Y
m (x z) f (x z) d (z), (x y) 2 , (7)
for all f 2 Mb ( F). Then Rb is a -order continuous positive projection
onto Mb (X ).
De nition 3.7 Let L be aDedekind -complete Riesz space with weak order
unit 0 < w 2 L and suppose that P : L ! L is a positive projection onto
the Riesz subspace K L with w 2 K. Give a measurable space (X )
and a - nite measure space (Y ), we will say that the product space
( F) =(X Y ) is a representation space for P if:
(i). ( F) is a representation space for (L w) with representation homomorphism
:b L ! L
(ii). there exists 0 m 2 M ( F) satisfying R m (x y) d (y) =1 for all
Y
x 2 X
(iii). if f 2 bL, then
Z
Y
m (x y) jf (x y)j d (y) < 1
for all x 2 X and the function Rf on , de ned by
Rf (x y) =
for all (x y) 2 , satis es Rf 2 bL
(iv). P ( f) = (Rf) for all f 2 b L.
Z
Y
m (x z) f (x z) d (z)
If, in addition, is aprobability measure and m (x y) =1for all (x y) 2 ,
then ( F) will be called a proper representation space for P .
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