## Operators defined by conditional expectations and random measures

##### Abstract

This study revolves around operators defined by conditional expectations
and operators generated by random measures.
Studies of operators in function spaces defined by conditional expectations
first appeared in the mid 1950's by S-T.C. Moy [22] and S. Sidak [26].
N. Kalton studied them in the setting of Lp-spaces 0 < p < 1 in [15, 131 and
in L1-spaces, [14], while W. Arveson [5] studied them in L2-spaces. Their
averaging properties were studied by P.G. Dodds and C.B. Huijsmans and
B. de Pagter in [7] and C.B. Huijsmans and B. de Pagter in [lo]. A. Lambert
[17] studied their relationship with multiplication operators in C*-modules.
It was shown by J.J. Grobler and B. de Pagter [8] that partial integral operators
that were studied A.S. Kalitvin et a1 in [2, 4, 3, 11, 121 and the special
cases of kernel operators that were, inter alia, studied by A.R. Schep in [25]
were special cases of conditional expectation operators.
On the other hand, operators generated by random measures or pseudo-integral
operators were studied by A. Sourour [28, 271 and L.W. Weis [29,30],
building on the studies of W. Arveson [5] and N. Kalton [14, 151, in the late
1970's and early 1980's.
In this thesis we extend the work of J.J. Grobler and B. de Pagter [8] on
Multiplication Conditional Expectation-representable (MCE-representable)
operators. We also generalize the result of A. Sourour [27] and show that
order continuous linear maps between ideals of almost everywhere finite
measurable functions on u-finite measure spaces are MCE-representable.
This fact enables us to easily deduce that sums and compositions of MCE-representable
operators are again MCE-representable operators. We also
show that operators generated by random measures are MCE-representable.
The first chapter gathers the definitions and introduces notions and concepts
that are used throughout. In particular, we introduce Riesz spaces and
operators therein, Riesz and Boolean homomorphisms, conditional expectation
operators, kernel and absolute T-kernel operators.
In Chapter 2 we look at MCE-operators where we give a definition different
from that given by J.J. Grobler and B. de Pagter in [8], but which we
show to be equivalent.
Chapter 3 involves random measures and operators generated by random
measures. We solve the problem (positively) that was posed by A. Sourour
in [28] about the relationship of the lattice properties of operators generated
by random measures and the lattice properties of their generating random
measures. We show that the total variation of a random signed measure
representing an order bounded operator T, it being the difference of two
random measures, is again a random measure and represents ITI.
We also show that the set of all operators generated by a random measure
is a band in the Riesz space of all order bounded operators.
In Chapter 4 we investigate the relationship between operators generated
by random measures and MCE-representable operators. It was shown by
A. Sourour in [28, 271 that every order bounded order continuous linear
operator acting between ideals of almost everywhere measurable functions is
generated by a random measure, provided that the measure spaces involved
are standard measure spaces. We prove an analogue of this theorem for
the general case where the underlying measure spaces are a-finite. We also,
in this general setting, prove that every order continuous linear operator is
MCE-representable. This rather surprising result enables us to easily show
that sums, products and compositions of MCE-representable operator are
again MCE-representable.
Key words: Riesz spaces, conditional expectations, multiplication conditional
expectation-representable operators, random measures.