nLab C-star-system

Idea

A $C^*$-dynamical system, or only $C^*$-system is a C-star-algebra together with an action of a group of automorphisms. In quantum mechanics as well as in AQFT the observables of the theory are self-adjoint operators of (a local net of) C-star-algebras, in this context the global gauge group of the theory is the maximal group of unitary operators that leave all observables invariant, the algebra and the gauge group form a $C^*$-system.

Definition

Definition

A $C^*$-system $(\mathcal{A}, \alpha_G)$ consists of a $C^*$-algebra $\mathcal{A}$, a locally compact group $G$ and a continuous homomorphism $\alpha$ of $G$ into the group $aut(\mathcal{A})$ of $*$-automorphisms of $\mathcal{A}$ equipped with the topology of pointwise convergence.

If the algebra is a $*$-algebra only, then some authors call it a $*$-system.

Sometimes the continuity condition is dropped entirely or replaced by some weaker assumption, therefore one should always check what – if any – continuity assumption an author makes.

Definition

The fixed point algebra of a $C^*$-system $(\mathcal{A}, \alpha_G)$ is $\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G \}$. If the fixed point algebra is trivial then $\alpha_G$ acts ergodically.

Definition

A state $\rho$ of the algebra $\mathcal{A}$ is an invariant state if

$$\rho (A) = \rho(\alpha_g A) \; \forall A \in \mathcal{A}, \; \forall g \in G.$$

Properties

Lemma

The set of invariant states is convex, weak-$*$ closed and weak-$*$ compact. (see operator topology).

References

• Hellmut Baumgärtel, Manfred Wollenberg: Causal nets of operator algebras. Berlin: Akademie Verlag 1992 (ZMATH entry)

• chapter 6 in Gerd Petersen, Pullback and pushout constructions in $C^\ast$-algebra theory, J. Funct. Analysis 167, 243–344 (1999) pdf

category: operator algebras