A -dynamical system, or only -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 -system.
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.
The fixed point algebra of a -system is . If the fixed point algebra is trivial then acts ergodically.
A state of the algebra is an invariant state if
The set of invariant states is convex, weak- closed and weak- compact. (see operator topology).
chapter 6 in Gerd Petersen, Pullback and pushout constructions in -algebra theory, J. Funct. Analysis 167, 243–344 (1999) pdf