A C *C^*-dynamical system, or only C *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 *C^*-system.



A C *C^*-system (๐’œ,ฮฑ G)(\mathcal{A}, \alpha_G) consists of a C *C^*-algebra ๐’œ\mathcal{A}, a locally compact group GG and a continuous homomorphism ฮฑ\alpha of GG into the group aut(๐’œ)aut(\mathcal{A}) of **-automorphisms of ๐’œ\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 โ€“ if any โ€“ continuity assumption an author makes.


The fixed point algebra of a C *C^*-system (๐’œ,ฮฑ G)(\mathcal{A}, \alpha_G) is {Aโˆˆ๐’œ:a gA=Aโˆ€gโˆˆG}\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G \}. If the fixed point algebra is trivial then ฮฑ G\alpha_G acts ergodically.


A state ฯ\rho of the algebra ๐’œ\mathcal{A} is an invariant state if

ฯ(A)=ฯ(ฮฑ gA)โˆ€Aโˆˆ๐’œ,โˆ€gโˆˆG. \rho (A) = \rho(\alpha_g A) \; \forall A \in \mathcal{A}, \; \forall g \in G.



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


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

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

Created on December 23, 2013 08:06:54 by Zoran ล koda (