### Abstract

Let F be the Fermion C-*-algebra and let F-infinity be its (Pedersen) Borel (*)-envelope. Then there exists a sigma-ideal, M, the meagre ideal, such that F-infinity/M is the regular (sigma)-completion of F; a monotone complete type III factor which has no normal states. Also, there exists a sigma-ideal, N, such that F-infinity/N can be identified with the generic dynamics factor; a monotone complete type III factor which has no normal states. It might be considered plausible that M = N. We show here that this is false. There exists a central projection q in F-infinity such that q is in M and 1 - q is in N. Hence F-infinity = qM target (1 - q)N.

Original language | English |
---|---|

Pages (from-to) | 365-370 |

Number of pages | 5 |

Journal | Quarterly Journal of Mathematics |

Volume | 53 |

DOIs | |

Publication status | Published - 2002 |

### Keywords

- C-STAR-ALGEBRAS
- GENERIC DYNAMICS

### Cite this

*Quarterly Journal of Mathematics*,

*53*, 365-370. https://doi.org/10.1093/qjmath/53.3.365

**Monotone completions of the Fermion algebra.** / Saito, Kazuyuki; Wright, John David Maitland.

Research output: Contribution to journal › Article

*Quarterly Journal of Mathematics*, vol. 53, pp. 365-370. https://doi.org/10.1093/qjmath/53.3.365

}

TY - JOUR

T1 - Monotone completions of the Fermion algebra

AU - Saito, Kazuyuki

AU - Wright, John David Maitland

PY - 2002

Y1 - 2002

N2 - Let F be the Fermion C-*-algebra and let F-infinity be its (Pedersen) Borel (*)-envelope. Then there exists a sigma-ideal, M, the meagre ideal, such that F-infinity/M is the regular (sigma)-completion of F; a monotone complete type III factor which has no normal states. Also, there exists a sigma-ideal, N, such that F-infinity/N can be identified with the generic dynamics factor; a monotone complete type III factor which has no normal states. It might be considered plausible that M = N. We show here that this is false. There exists a central projection q in F-infinity such that q is in M and 1 - q is in N. Hence F-infinity = qM target (1 - q)N.

AB - Let F be the Fermion C-*-algebra and let F-infinity be its (Pedersen) Borel (*)-envelope. Then there exists a sigma-ideal, M, the meagre ideal, such that F-infinity/M is the regular (sigma)-completion of F; a monotone complete type III factor which has no normal states. Also, there exists a sigma-ideal, N, such that F-infinity/N can be identified with the generic dynamics factor; a monotone complete type III factor which has no normal states. It might be considered plausible that M = N. We show here that this is false. There exists a central projection q in F-infinity such that q is in M and 1 - q is in N. Hence F-infinity = qM target (1 - q)N.

KW - C-STAR-ALGEBRAS

KW - GENERIC DYNAMICS

U2 - 10.1093/qjmath/53.3.365

DO - 10.1093/qjmath/53.3.365

M3 - Article

VL - 53

SP - 365

EP - 370

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

ER -