nLab
C-star-system

Contents

Idea

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.

Definition

Definition

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

Definition

The fixed point algebra of a C *C^*-system (π’œ,Ξ± G)(\mathcal{A}, \alpha_G) is {Aβˆˆπ’œ:a gA=Aβˆ€g∈G}\{ A \in \mathcal{A}: a_g A = A \; \forall \; g \in G \}. If the fixed point algebra is trivial then Ξ± G\alpha_G acts ergodically.

Definition

A state ρ\rho of the algebra π’œ\mathcal{A} is an invariant state if

ρ(A)=ρ(Ξ± gA)βˆ€Aβˆˆπ’œ,βˆ€g∈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 BaumgΓ€rtel, 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, 243–344 (1999) pdf

Created on December 23, 2013 08:06:54 by Zoran Ε koda (95.168.114.9)