Conditional expectation

Let $X$ be a sample space, $F$ be a sigma algebra and $G$ a sub-sigma algebra of $F$.

Conditional expectation is a projection of random variables, $E: L^2(X,F) → L^2(X,G)$. The operator can be extended to $L^1(X,F)$ and is a Markov operator.

 

How is it related to conditional probability?

 

Let $A$ be an event, $\{B_i\}$ be a decomposition of $X$ and $G$ be the sub-sigma algebra generated by $\{B_i\}$. The conditional expectation of the indicator function $1_A$ is a linear combination of $1_{B_i}$. Here the coefficients are the conditional probabilities $P(A|B_i)$! In this sense, the conditional expectation packages all the conditional probabilities together.

 

More generally, given a $C^*$-subalgebra $B$ of a $C^*$-algebra $A$ having the same unit, conditional expectation is a positive, surjective, unital operator $T:A→B$ such that $T^2 = T$.

 

Rotation algebra is the universal $C^*$-algebra generated by unitaries $u$ and $v$ with $uv = e^{2 \pi i \theta} vu$. Cuntz algebra is the universal $C^*$-algebra generated by isometries $S_i$ where summation of $\sum_i S_i S_i^* = 1$.

 

Conditional expectation is used to prove the simplicity of rotation algebra and Cuntz algebra.

 

References:

* Functional Analysis for Probability and Stochastic Processes, Adam Bobrowski

* https://www.math.uni-sb.de/ag/speicher/weber/ISem24/ISem24LectureNotes.pdf