$C^*$-index of observable algebra in the field algebra determined by a normal group

TL;DR

This paper investigates the $C^*$-index of the observable algebra within the field algebra of $G$-spin models, constructed via a normal subgroup of a finite group, revealing new algebraic invariants in quantum algebraic structures.

Contribution

It introduces a concrete construction of a $D(H;G)$-invariant subalgebra and computes its $C^*$-index using a quasi-basis of the conditional expectation.

Findings

Construction of a $D(H;G)$-invariant subalgebra ${ m extbf{A}}_{(H,G)}$

Explicit calculation of the $C^*$-index of the conditional expectation

Establishment of algebraic invariants related to normal subgroups in quantum models

Abstract

Let $G$ be a finite group and $H$ a normal subgroup. $D(H;G)$ is the crossed product of $C(H)$ and ${\Bbb C}G$ which is only a subalgebra of $D(G)$, the quantum double of $G$. One can construct a $C^*$-subalgebra ${\mathcal{F}}_{_H}$ of the field algebra $\mathcal{F}$ of $G$-spin models, such that ${\mathcal{F}}_{_H}$ is a $D(H;G)$-module algebra. The concrete construction of $D(H;G)$-invariant subalgebra ${\mathcal{A}}_{_{(H,G)}}$ of ${\mathcal{F}}_{_H}$ is given. By constructing the quasi-basis of conditional expectation $z_{_H}$ of ${\mathcal{F}}_{_H}$ onto ${\mathcal{A}}_{_{(H,G)}}$, the $C^*$-index of $z_{_H}$ is given.