## Abstract

In 1955, H. Dye defined certain projections of a C*-matrix algebraby [formula] which was used to show that in the case of factors not of type I_{2n}, the unitarygroup determines the algebraic type of that factor. We study these projectionsand we show that in M_{2}(C), the set of such projections includes all the projections. For infinite C*-algebra A, having a system of matrix units, we haveA ≃ M_{n}(A). M. Leen proved that in a simple, purely infinite C*-algebra A, the *-symmetries generate u_{0}(A). Assuming K_{1}(A) is trivial, we revise Leen’sproof and we use the same construction to show that any unitary close to theunity can be written as a product of eleven *-symmetries, eight of such are ofthe form 1 − 2P_{i, j} (ω), ω ∈ u (A). In simple, unital purely infinite C*-algebrashaving trivial K_{1}-group, we prove that all P_{i, j} (ω) have trivial K_{0}-class. Consequently, we prove that every unitary of O_{n} can be written as a finite productof *-symmetries, of which a multiple of eight are conjugate as group elements.

Original language | English |
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Pages (from-to) | 144-154 |

Number of pages | 11 |

Journal | Annals of Functional Analysis |

Volume | 3 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 |

## Keywords

- C*-algebras
- K-class
- Matrix projection

## ASJC Scopus subject areas

- Analysis
- Anatomy
- Algebra and Number Theory