### Abstract:

Lambert (1999) used conditional expectation to contruct certain operator spaces. The conditional expectation induces a Hilbert module structure on these spaces. Application is made of Kasparov's theorem to examine certain multiplier and compact algebras associated with the module structure. This examination is related to the classic results on characterizations of conditional expectation type-operators.<br><br> Let L be an ideal of measurable functions and let S : L → L be a positive, order continuous linear projection onto a sublattice, which preserves one. We can characterize the conditional expectation in terms of S : L → L. We follow the pattern followed by Dodds, Huijsmans and De Pagter (1990) in order to extend this result. We first let 51 be strictly positive, and then decompose L as a direct sum of the band generated by Si and its disjoint complement. Finally if S is not necessarily strictly positive, we first decompose L into the carrier band of S and the absolute kernel of S and then apply the preceding arguments to arrive at a matrix representation of S.<br><br> By means of certain results on averaging contractions, due to Douglas (1965) and Seever (1966), we extend the first characterization of conditional expectation (given above) to a positive, order continuous projection, S : L → L, which preserves one, and such that the dual, S<sup>1</sup> has an extention to L<sup>1</sup> + L<sup>∞</sup>(Ω,Σ, μ) which preserves one. In order to proof this, one needs the fact that if S can be characterized in terms of conditional expectation, then the same is valid for S<sup>1</sup> : L<sup>1</sup> → L<sup>1</sup>. The proof of this result, which enable us to give the final characterization of conditional expectation, is also given.