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Derivations on ideals in commutative AW*-algebras

Abstract

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

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Correspondence to V. I. Chilin.

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Original Russian Text © V.I. Chilin and G.B. Levitina, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 63–88.

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Chilin, V.I., Levitina, G.B. Derivations on ideals in commutative AW*-algebras. Sib. Adv. Math. 24, 26–42 (2014). https://doi.org/10.3103/S1055134414010040

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Keywords

  • boolean algebra
  • commutative AW*-algebra
  • ideal
  • derivation