### Abstract

Let A be a semisimple Danach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Danach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation expansion formula on zero products.

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

Pages (from-to) | 193-202 |

Number of pages | 10 |

Journal | Studia Mathematica |

Volume | 190 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2009 Mar 18 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Studia Mathematica*,

*190*(2), 193-202. https://doi.org/10.4064/sm190-2-7

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*Studia Mathematica*, vol. 190, no. 2, pp. 193-202. https://doi.org/10.4064/sm190-2-7

**Partially defined σ-derivations on semisimple Banach algebras.** / Lee, Tsiu Kwen; Liu, Cheng-Kai.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Partially defined σ-derivations on semisimple Banach algebras

AU - Lee, Tsiu Kwen

AU - Liu, Cheng-Kai

PY - 2009/3/18

Y1 - 2009/3/18

N2 - Let A be a semisimple Danach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Danach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation expansion formula on zero products.

AB - Let A be a semisimple Danach algebra with a linear automorphism σ and let δ: I → A be a σ-derivation, where I is an ideal of A. Then Φ(δ)(I ∩ σ(I)) = 0, where Φ(δ) is the separating space of δ. As a consequence, if I is an essential ideal then the σ-derivation δ is closable. In a prime C*-algebra, we show that every σ-derivation defined on a nonzero ideal is continuous. Finally, any linear map on a prime semisimple Danach algebra with nontrivial idempotents is continuous if it satisfies the σ-derivation expansion formula on zero products.

UR - http://www.scopus.com/inward/record.url?scp=62149148460&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=62149148460&partnerID=8YFLogxK

U2 - 10.4064/sm190-2-7

DO - 10.4064/sm190-2-7

M3 - Article

AN - SCOPUS:62149148460

VL - 190

SP - 193

EP - 202

JO - Studia Mathematica

JF - Studia Mathematica

SN - 0039-3223

IS - 2

ER -