# Contents

## 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 $\left(𝒜,{\alpha }_{G}\right)$ consists of a ${C}^{*}$-algebra $𝒜$, a locally compact group $G$ and a continuous homomorphism $\alpha$ of $G$ into the group $\mathrm{aut}\left(𝒜\right)$ 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 $\left(𝒜,{\alpha }_{G}\right)$ is $\left\{A\in 𝒜:{a}_{g}A=A\phantom{\rule{thickmathspace}{0ex}}\forall \phantom{\rule{thickmathspace}{0ex}}g\in G\right\}$. 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

$\rho \left(A\right)=\rho \left({\alpha }_{g}A\right)\phantom{\rule{thickmathspace}{0ex}}\forall A\in 𝒜,\phantom{\rule{thickmathspace}{0ex}}\forall g\in G.$\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

Revised on July 31, 2011 00:11:08 by Urs Schreiber (82.113.99.54)