# Partially defined σ-derivations on semisimple Banach algebras

Tsiu Kwen Lee, Cheng-Kai Liu

Research output: Contribution to journalArticle

4 Citations (Scopus)

### 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 193-202 10 Studia Mathematica 190 2 https://doi.org/10.4064/sm190-2-7 Published - 2009 Mar 18

Banach algebra
Semisimple
Algebra
Linear map
Idempotent
Automorphism
C*-algebra
Zero

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

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title = "Partially defined σ-derivations on semisimple Banach algebras",
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.",
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In: Studia Mathematica, Vol. 190, No. 2, 18.03.2009, p. 193-202.

Research output: Contribution to journalArticle

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T1 - Partially defined σ-derivations on semisimple Banach algebras

AU - Lee, Tsiu Kwen

AU - Liu, Cheng-Kai

PY - 2009/3/18

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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.

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