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 Mat. Tr., 2013, Volume 16, Number 1, Pages 63–88 (Mi mt250)

Derivations on ideals in commutative $AW^*$-algebras

G. B. Levitina, V. I. Chilin

National University of Uzbekistan, Faculty of Mathematics and Mechanics, Tashkent, Uzbekistan

Abstract: Let $\mathcal A$ be a commutative $AW^*$-algebra.We denote by $S(\mathcal A)$ the $*$-algebra of measurable operators that are affiliated with $\mathcal A$. For an ideal $\mathcal I$ in $\mathcal A$, let $s(\mathcal I)$ denote the support of $\mathcal I$. Let $\mathbb Y$ be a solid linear subspace in $S(\mathcal A)$. We find necessary and sufficient conditions for existence of nonzero band preserving derivations from $\mathcal I$ to $\mathbb Y$. We prove that no nonzero band preserving derivation from $\mathcal I$ to $\mathbb Y$ exists if either $\mathbb Y\subset\mathcal A$ or $\mathbb Y$ is a quasi-normed solid space. We also show that a nonzero band preserving derivation from $\mathcal I$ to $S(\mathcal A)$ exists if and only if the boolean algebra of projections in the $AW^*$-algebra $s(\mathcal I)\mathcal A$ is not $\sigma$-distributive.

Key words: Boolean algebra, commutative $AW^*$-algebra, ideal, derivation.

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English version:
Siberian Advances in Mathematics, 2014, 24:1, 26–42

Bibliographic databases:

UDC: 517.98

Citation: G. B. Levitina, V. I. Chilin, “Derivations on ideals in commutative $AW^*$-algebras”, Mat. Tr., 16:1 (2013), 63–88; Siberian Adv. Math., 24:1 (2014), 26–42

Citation in format AMSBIB
\Bibitem{LevChi13} \by G.~B.~Levitina, V.~I.~Chilin \paper Derivations on ideals in commutative $AW^*$-algebras \jour Mat. Tr. \yr 2013 \vol 16 \issue 1 \pages 63--88 \mathnet{http://mi.mathnet.ru/mt250} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3156674} \transl \jour Siberian Adv. Math. \yr 2014 \vol 24 \issue 1 \pages 26--42 \crossref{https://doi.org/10.3103/S1055134414010040} 

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This publication is cited in the following articles:
1. A. A. Alimov, V. I. Chilin, “Differentsirovaniya so znacheniyami v idealnykh $F$-prostranstvakh izmerimykh funktsii”, Vladikavk. matem. zhurn., 20:1 (2018), 21–29
2. A. F. Ber, V. I. Chilin, F. A. Sukochev, “Derivations on Banach $*$-ideals in von Neumann algebras”, Vladikavk. matem. zhurn., 20:2 (2018), 23–28
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