Hilbert C*-systems for actions of the circle group

TL;DR

This paper constructs Hilbert systems for circle group actions using subgroups of implementable Bogoljubov unitaries in Fock representations of the Fermion algebra, providing explicit examples and calculations.

Contribution

It introduces a method to construct Hilbert systems for circle group actions via Bogoljubov unitaries in Fermion algebra representations, with explicit center calculations.

Findings

The group of T-valued functions is isomorphic to the stabilizer of the algebra.

Explicit examples where the center of the fixed point algebra is calculated.

Construction of Hilbert systems using selfdual framework data.

Abstract

The paper contains constructions of Hilbert systems for the action of the circle group $T$ using subgroups of implementable Bogoljubov unitaries w.r.t. Fock representations of the Fermion algebra for suitable data of the selfdual framework: ${\cal H}$ is the reference Hilbert space, $\Gamma$ the conjugation and $P$ a basis projection on ${\cal H}.$ The group $C({spec} {\cal Z}\to T)$ of $T$-valued functions on ${spec} {\cal Z}$ turns out to be isomorphic to the stabilizer of ${\cal A}$. In particular, examples are presented where the center ${\cal Z}$ of the fixed point algebra ${\cal A}$ can be calculated explicitly.