Hilbert \(C^*\)-systems for actions of the circle group (Q5945639)

The authors present various constructions of Hilbert systems \(({\mathcal F}, \alpha_T)\) where \(T\) is the circle group. The direct approach consists of building a \(C^*\)-algebra \({\mathcal F}\) of operators acting on \(L^2(T, K)\) where \(K\) is a Hilbert space. Then \({\mathcal F}\) is an arbitrary \(C^*\)-algebra containing the bilateral shift \(V\) and the left regular representation \(U_\zeta, \zeta \in T\). The automorphism group acting on \({\mathcal F}\) consists of \(\text{Ad} U_\zeta |{\mathcal F}\), \(\zeta \in T\). By applying quasi-free fermion quantization to the above procedure, one gets Hilbert systems for which the representation of \(T\) defining the automorphism group acting on \({\mathcal F}\) has nonnegative spectrum. Moreover, the authors present a detailed procedure leading to Hilbert systems for which the center of the fixed point algebra can be calculated explicitly. An example using the implementation of the loop group of such Hilbert systems is given.