Random Variables
We give a treatment of the statement that given a probability space \((\Omega, \Sigma, \mathbb{P})\), the algebra of affiliated operators to \((L^{\infty}(\Omega), E[\cdot])\) is all real valued random variables.
Classical Definition
Let \((\Omega, \Sigma, \mathbb{P})\) be a probability space and \((\mathbb{R}, \mathcal{B})\) be the usual real line with its Borel \(\sigma\)-algebra.
We say a function \(X:\Omega \to \mathbb{R}\) is a real-valued random variable if \(X\) is measurable. That is, symbolically,
\[\forall B \in \mathcal{B} \implies X^{\leftarrow}(B) \in \Sigma\]So, let us answer a very simple question: can we describe the set of all random variables, \(\{X \in\mathbb{R}^{\Omega} : X \text{ is measurable}\}\)?
\(W^{\star}\) algebras (von Neumann Algebras)
Fix a Hilbert space \(\mathcal{H}\), and let \(B(\mathcal{H})\) be the set of all bounded operators \(T: \mathcal{H} \to \mathcal{H}\).
Definition: A \(W^{\star}\) algebra acting on \(\mathcal{H}\) is a sub-algebra \(\mathcal{M} \subset B(\mathcal{H})\) such that \(\mathcal{M}\) is
- unital
- self-adjoint
- strong-operator topology closed
(see von Neumann’s bicommuntant theorem for equivalents)
Reminder: the strong-operator topology is generated by the subbase \(\{B(T,h, \epsilon): T\in B(\mathcal{H}), h\in \mathcal{H}, \epsilon >0\}\) where \(B(T,h,\epsilon) = \{S \in B(\mathcal{H} : ||(T-S)x|| < \epsilon\}\) where \(||\cdot||\) is the usual operator norm.
Definition: Given a \(W^{\star}\) algebra, \(\mathcal{M}\), a linear functional \(\tau\) is called a state on \(\mathcal{M}\)
- \[x\geq 0 \implies \tau(x) \geq 0\]
- \[\tau(1) = 1\]
Definition: A state \(\tau\) on \(\mathcal{M}\) is called tracial if forall \(x,y \in \mathcal{M}\)
\[\tau(xy) = \tau(yx)\]