nLab C-star dynamical system

Idea

A $C^*$-dynamical system (or $C^*$-system, for short) is a $C^\ast algebra$ continuously acted upon by a topological group of $\ast$-automorphisms.

In quantum mechanics and generally in AQFT, where quantum observables of the theory are self-adjoint operators of (a local net of) $C^\ast$-algebras, the global gauge group of the theory is the maximal group of unitary operators that leave all observables invariant, hence algebra of quantum observables equipped with this action of the global 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 $star$-algebra only, then some authors call it a $\ast$-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 (cf. operator topology).

Related entries

• net of $C^\ast$ systems

References

• Hellmut Baumgärtel, Manfred Wollenberg; chapter 6 of: Causal nets of operator algebras — Mathematical aspects of algebraic quantum field theory, Mathematische Lehrbücher und Monographien. II. Abteilung: Mathematische Monographien 80, Akademie Verlag (1992) [zbmath:0749.46038]

• Gert K. Pedersen; chapter 6 in: Pullback and pushout constructions in $C^\ast$-algebra theory, J. Funct. Analysis 167 (1999) 243–344 [doi:10.1006/jfan.1999.3456, pdf]

category: operator algebras