A Modified Approach to the Doplicher/Roberts Theorem on the Construction of the Field Algebra and the Symmetry Group in Superselection Theory
Hellmut Baumgärtel · Reviews in Mathematical Physics · 1997
Let [Formula: see text] be a unital C*-algebra with trivial center [Formula: see text]. Let [Formula: see text] denote a tensorial category of unital endomorphisms of [Formula: see text] equipped with several properties to be explained in the text. Doplicher and Roberts have shown, among other things, that there is a C*-algebra [Formula: see text] and a compact group [Formula: see text] of automorphisms of ℱ such that ℱ is a Hilbert C*-system over [Formula: see text] w.r.t. [Formula: see text], where [Formula: see text] is the fixed point algebra w.r.t. [Formula: see text], [Formula: see text] and the objects [Formula: see text] are characterized as the canonical endomorphisms of certain algebraic [Formula: see text]-invariant Hilbert spaces ℋρ⊂ℱ, see Doplicher/Roberts [1, 2, 3]. The starting point of the approach presented in this paper to point out the mentioned result is an [Formula: see text]-leftmodule ℱ0:={∑ρ,jAρ,jΦρ,j}. ρ runs through a full system of irreducible and mutually disjoint objects of [Formula: see text], j=1,2,…,d(ρ), where d(ρ) denotes the statistical dimension of [Formula: see text] is an orthonormal basis of a d(ρ)-dimensional Hilbert space. The system {Φρj}ρj forms a leftmodule basis of ℱ0, the coefficients Aρj are members of [Formula: see text]. The strategy is to equip successively ℱ0 with a bimodule structure, a product and a *-structure and finally with a C*-norm ||.||*. The symmetry group [Formula: see text] appears as the group of all automorphisms of the *-algebra ℱ0 leaving the [Formula: see text]-scalar product [Formula: see text] invariant, where F=∑ρ,jAρjΦρj, G=∑ρ,jBρjΦρj. The field algebra is then given by ℱ:= clo ||.||*ℱ0.