Characterization of conditional expectation operators for Banach-valued functions

Abstract:

Let $B$ be a Banach space. It is well known that for every probability space $(\Omega , \mathcal {A}, P)$ and every sub-$\sigma$-field $\mathcal {A}_0 \subset \mathcal {A}$ there exists a conditional expectation operator $P^{\mathcal {A}_0}X$ for $B$-valued $P$-integrable functions. This operator maps the space ${L_p}(\Omega , \mathcal {A}, P, B)$ into itself for each $p \geqslant 1$. The operator is linear, idempotent, constant preserving and contractive in ${L_p}$. For $B = {\mathbf {R}}$ and $p \ne 2$ these conditions characterize a conditional expectation operator. It turns out that in general these properties characterize conditional expectation operators for Banach-valued functions only for $p = 1$ and strictly convex Banach spaces.