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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On derivations annihilating a maximal abelian subalgebra


Author: Geoffrey L. Price
Journal: Trans. Amer. Math. Soc. 290 (1985), 843-850
MSC: Primary 46L40; Secondary 46L55
MathSciNet review: 792832
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Abstract: Let $ \mathcal{A}$ be an $ {\text{AF}}\;{C^\ast}$-algebra, and let $ \delta $ be a closed $ \ast $-derivation which annihilates the maximal abelian subalgebra $ \mathcal{C}$ of diagonal elements of $ \mathcal{A}$. Then we show that $ \delta $ generates an approximately inner $ {C^\ast}$-dynamics on $ \mathcal{A}$, and that $ \delta $ is a commutative $ \ast $-derivation. Any two closed $ \ast$-derivations vanishing on $ \mathcal{C}$ are shown to be strongly commuting. More generally, if $ \delta $ is a semiderivation on $ \mathcal{A}$ which vanishes on $ \mathcal{C}$, we prove that $ \delta $ is a generator of a semigroup of strongly positive contractions of $ \mathcal{A}$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0792832-1
PII: S 0002-9947(1985)0792832-1
Keywords: Derivation, $ {\text{AF}}$-algebra, maximal abelian subalgebra, approximately inner, generator
Article copyright: © Copyright 1985 American Mathematical Society