nLab
C-star-system

Idea

A $C^{*}$-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 $(𝒜,{\alpha }_G)$ consists of a $C^{*}$-algebra $𝒜$, a locally compact group $G$ and a continuous homomorphism $\alpha$ of $G$ into the group $\mathrm{aut}(𝒜)$ of $*$-automorphisms of $𝒜$ 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 $(𝒜,{\alpha }_G)$ is $\{A\in 𝒜:a_{g}A=A\forall g\in G\}$. If the fixed point algebra is trivial then ${\alpha }_G$ acts ergodically.

Definition

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

$$\rho (A)=\rho ({\alpha }_{g}A)\forall A\in 𝒜,\forall g\in G.$$\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)