PhyslibAlpha.ProbabilisticTheory.WStarAlgebra.Basic

For a $C^*$-algebra $A$ equipped with a $W^*$-algebra structure (providing an isometric identification of $A$ with the dual of a predual space $A_*$), the weak-$\star$ topology is the topology on $A$ induced by its pairing with $A_*$. It is the coarsest topology such that for every element $\xi \in A_*$, the linear map $A \to \mathbb{C}$ defined by $a \mapsto \xi(a)$ is continuous.

For a $W^*$-algebra $A$, the weak-$\star$ topology (the topology induced by the pairing with its predual $A_*$) is coarser than the topology induced by the norm $\|\cdot\|$ on $A$. That is, every open set in the weak-$\star$ topology is also an open set in the norm topology.

Given an element $\xi$ in the predual $A_*$ of a $W^*$-algebra $A$, this definition provides the corresponding continuous linear functional on $A$. For any $a \in A$, the value is given by the natural pairing $\langle a, \xi \rangle$, where $A$ is identified as the dual of its predual.

Let $A$ be a $W^*$-algebra and $A_*$ its predual. For any element $\xi \in A_*$ and any element $a \in A$, the value of the continuous linear functional induced by $\xi$ on $A$ is given by the evaluation of $a$ on $\xi$ via the canonical isometric identification $A \cong (A_*)^*$. That is, $\text{predualPairing}(\xi)(a) = \langle a, \xi \rangle$.

Let $A$ be a $W^*$-algebra and $A_*$ its predual. For any element $a \in A$ and $\xi \in A_*$, the pairing $\langle a, \xi \rangle$ (representing the evaluation of the linear functional $\xi$ at $a$) satisfies the norm inequality: $$\|\langle a, \xi \rangle\| \leq \|a\| \cdot \|\xi\|$$ where $\|\cdot\|$ denotes the norm in $A$, its predual $A_*$, and the complex numbers $\mathbb{C}$ respectively.

Let $A$ be a $W^*$-algebra and $A_*$ be its predual. For any element $\xi \in A_*$, the linear functional on $A$ defined by the pairing $\langle \cdot, \xi \rangle : A \to \mathbb{C}$ (where $\langle a, \xi \rangle$ is the value of the functional $\xi$ at $a$) is continuous with respect to the weak-$\star$ topology on $A$ and the standard topology on $\mathbb{C}$.

Let $A$ be a $W^*$-algebra with predual $A_*$. For any two elements $a, b \in A$, if $\langle a, \xi \rangle = \langle b, \xi \rangle$ for all $\xi \in A_*$, then $a = b$. Here, $\langle a, \xi \rangle$ denotes the natural pairing between an element of the algebra and an element of its predual.

Let $A$ be a $W^*$-algebra and $A_*$ be its predual. Let $f: \alpha \to A$ be a function and $l$ be a filter on $\alpha$. Then $f$ converges to $a \in A$ in the weak-$\star$ topology if and only if for every element $\xi \in A_*$, the complex-valued function $i \mapsto \langle f(i), \xi \rangle$ converges to $\langle a, \xi \rangle$ along the filter $l$. Here, $\langle \cdot, \xi \rangle$ denotes the continuous linear functional on $A$ induced by the predual element $\xi$.

For a $W^*$-algebra $A$, this instance allows a normal state $\omega$ to be treated as a function from the algebra to the complex numbers, mapping $a \in A$ to $\omega(a) \in \mathbb{C}$.

Let $A$ be a $W^*$-algebra and $\omega$ be a normal state on $A$. For any element $a \in A$, the value of the underlying state $\omega.\text{toState}$ applied to $a$ is equal to the value of $\omega$ applied to $a$, i.e., $\omega.\text{toState}(a) = \omega(a)$.

Let $A$ be a $W^*$-algebra and let $\omega$ be a normal state on $A$. Then $\omega$ is continuous as a mapping from $A$ to $\mathbb{C}$ with respect to the norm topology. This follows because a normal state is defined to be continuous with respect to the weak-$\star$ topology, and the weak-$\star$ topology is coarser than the norm topology.