C*-Dynamical Systems

Group Actions and C*-Dynamical Systems

Throughout, we assume $G$ is a countable group equipped with the discrete topology.

Recall the definition of a group action:

A set $X$ equipped with an action of $G$ is called a $G$-set.

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If $X$ is a topological space, we say that $G$ acts continuously if the action map $G\times X\to X$ is continuous.

e.g.

• Let $G=(\mathbb{R},+)$ and $X={\mathbb{R}}^n$. Consider an initial value problem given by $\dot{x}(t)=F(x(t))$ for a sufficiently nice function $F:{\mathbb{R}}^n\to {\mathbb{R}}^n$. For any $z\in {\mathbb{R}}^n$, let $x_z(t)$ be the unique solution with initial condition $x_z(0)=z$. We can define an action of $\mathbb{R}$ on ${\mathbb{R}}^n$ by $t\cdot z:=x_z(t)$.

  • $0\cdot z=x_z(0)=z$.
  • $s\cdot (t\cdot z)=s\cdot x_z(t)$. The solution starting at $x_z(t)$ is just a time-shift of the original solution, i.e., $x_{x_z(t)}(s)=x_z(t+s)$. Thus, $s\cdot (t\cdot z)=x_z(t+s)=(t+s)\cdot z$.

• Let be a locally compact topological space and let $h\in \text{Homeo}(X)$ be a homeomorphism. The group $G=\mathbb{Z}$ acts continuously on via $n\cdot x:=h^n(x)$ for $n\in \mathbb{Z}$ and $x\in X$.
  

A group action on a space $X$ induces a natural action on the C*-algebra of continuous functions on $X$.

Proposition

Let $G$ be a group acting continuously on a locally compact space $X$. This induces a map $\alpha :G\to \text{Aut}(C_0(X))$ given by
$({\alpha }_s(f))(x)=f(s^{-1}\cdot x),s\in G,x\in X,f\in C_0(X)$
This map $\alpha$ is a group homomorphism.

C*-Dynamical System

A C*-dynamical system is a triple $(\mathcal{A},G,\alpha )$, where $\mathcal{A}$ is a C*-algebra, $G$ is a group, and $\alpha :G\to \text{Aut}(\mathcal{A})$ is a homomorphism. The homomorphism is often required to be continuous in an appropriate sense.

Proposition

If $(C_0(X),G,\alpha )$ is a C*-dynamical system, then there exists a continuous action of $G$ on $X$ that induces $\alpha$.

Proof Sketch For each $s\in G$, ${\alpha }_s$ is an automorphism of $C_0(X)$. By the Gelfand-Naimark theorem, there exists a unique homeomorphism $h_s:X\to X$ such that ${\alpha }_s(f)=f\circ h_s^{-1}$ for all $f\in C_0(X)$. The homomorphism property ${\alpha }_{st}={\alpha }_s\circ {\alpha }_t$ implies $h_{st}=h_s\circ h_t$, defining a group action $s\cdot x:=h_s(x)$.

Covariant Representations

Covariant Representation

Let $(\mathcal{A},G,\alpha )$ be a C*-dynamical system. A covariant representation of $(\mathcal{A},G,\alpha )$ on a Hilbert space \newcommand{\H}{\mathcal{H}}\H is a pair $(\pi ,U)$ where:

1. $\pi :\mathcal{A}\to B(\stackrel{˝}{)}$ is a $\ast$-representation of $\mathcal{A}$ on \H.
2. $U:G\to U(\stackrel{˝}{)}$ is a unitary representation of $G$ on \H.
3. They satisfy the covariance condition: $\pi ({\alpha }_s(a))=U_s\pi (a)U_s^{\ast }$ for all $a\in \mathcal{A}$ and $s\in G$.
   

e.g. Let $(C_0(G),G,L)$ be the dynamical system where $G$ acts on itself by left multiplication, and $L$ is the action on functions by left translation: $(L_{s}f)(t)=f(s^{-1}t)$. Define a representation $\pi$ of $C_0(G)$ on ${\ell }^2(G)$ by multiplication operators: $(\pi (f)\xi )(s)=f(s)\xi (s)\text{for }f\in C_0(G),\xi \in {\ell }^2(G)$Let $\lambda$ be the left regular representation of $G$ on ${\ell }^2(G)$: $({\lambda }_s\xi )(t)=\xi (s^{-1}t)$Then $(\pi ,\lambda )$ is a covariant representation.

The Regular Representation Construction

Given any C*-dynamical system $(\mathcal{A},G,\alpha )$ and a representation ${\pi }_0:\mathcal{A}\to B(\stackrel{˝}{)}$, one can construct a covariant representation on the larger Hilbert space ${\ell }^2(G,\stackrel{˝}{)}$. It consists of functions \xi\colon G \to \H such that ${\sum }_{t\in G}\Vert \xi (t){\Vert }_H^2<\infty$. The inner product is $⟨\xi ,\eta ⟩={\sum }_{t\in G}⟨\xi (t),\eta (t){⟩}_H$.

1. Unitary Representation $\lambda$: Define $\lambda :G\to \mathcal{U}({\ell }^2(G,H))$ by $({\lambda }_s\xi )(t)=\xi (s^{-1}t)$This is a unitary representation of $G$.
2. Algebra Representation $\Pi$: Define $\Pi :\mathcal{A}\to B({\ell }^2(G,H))$ by $(\Pi (a)\xi )(t)={\pi }_0({\alpha }_{t^{-1}}(a))\xi (t)$
   

Theorem

The pair $(\Pi ,\lambda )$ constructed above is a covariant representation of $(\mathcal{A},G,\alpha )$.

Proof We must check that ${\lambda }_s\Pi (a){\lambda }_s^{\ast }=\Pi ({\alpha }_s(a))$. For any $\xi \in {\ell }^2(G,H)$ and $t\in G$: $\begin{array}{rl}({\lambda }_s\Pi (a){\lambda }_s^{\ast }\xi )(t)& =(\Pi (a){\lambda }_s^{\ast }\xi )(s^{-1}t)\\ & ={\pi }_0({\alpha }_{(s^{-1}t{)}^{-1}}(a))({\lambda }_s^{\ast }\xi )(s^{-1}t)\\ & ={\pi }_0({\alpha }_{t^{-1}s}(a))\xi (s(s^{-1}t))\\ & ={\pi }_0({\alpha }_{t^{-1}s}(a))\xi (t)\\ & ={\pi }_0({\alpha }_{t^{-1}}({\alpha }_s(a)))\xi (t)\\ & =(\Pi ({\alpha }_s(a))\xi )(t)\end{array}$The calculation follows from the definitions of $\Pi$, $\lambda$, and the homomorphism property of $\alpha$. Thus, the covariance condition holds. $\square$