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.
Similar content being viewed by others
References
S. Albeverio, Sh. A. Ayupov, and K. K. Kudaybergenov, “Structure of derivations on various algebras of measurable operators for type I von Neumann algebras,” J. Funct. Anal. 256(9), 2917–2943 (2009).
A. F. Ber, V. I. Chilin, and F. A. Sukochev, “Non-trivial derivations on commutative regular algebras,” Extracta Math. 21(2), 107–147 (2006).
A. F. Ber and F. A. Sukochev, “Derivations in the Banach ideals of τ -compact operators,” (ArXiv:1204.4052 v1 [math.OA] 2012).
A. F. Ber, F. A. Sukochev, and V. I. Chilin, “Derivations in commutative regular algebras,” Math. Notes 75(3), 418–419 (2004) [Mat. Zametki 75 (3), 453–454 (2004)].
S. K. Berberian, Baer *-Rings (Springer-Verlag, New York-Berlin, 1972).
V. I. Chilin, “Partially ordered Baer involutive algebras,” J. SovietMath. 37, 1449–1472 (1987) [Itogi Nauki i Techniki. Current Problems in Mathematics. Newest Results 27, 99–128 (1985)].
A. E. Gutman, A. G. Kusraev, and S. S. Kutateladze, “The Wickstead problem,” Sib. Èlectron. Mat. Izv. 5, 293–333 (2008).
L. V. Kantorovich and G. P. Akilov, Functional Analysis (Pergamon Press, Oxford, 1982) [Functional Analysis (Nauka, Moscow, 1984].
S. G. Kreĭn, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (Amer. Math. Soc., Providence, RI, 1982). [Interpolation of Linear Operators (Nauka, Moscow, 1978)].
A. G. Kusraev, Vector Duality and Its Applications (Nauka, Novosibirsk, 1985) [in Russian].
A. G. Kusraev, Dominated Operators (Kluwer, Dordrecht, 2000) [Dominated Operators (Nauka, Moscow, 2003)].
A. G. Kusraev, “Automorphisms and derivations on a universally complete complex f-algebra,” Sibirsk. Mat. Zh. 47(1), 97–107 (2006) [Siberian Math. J. 47 (1), 77–85 (2006)].
D. Olesen, “Derivations of AW*-algebras are inner,” Pacific J. Math. 53(1), 555–561 (1974).
S. Sakai, C*-Algebras and W*-Algebras (Springer-Verlag, Berlin-Heidelberg-New York, 1971).
I. E. Segal, “A non-commutative extension of abstract integration,” Ann. of Math. (2) 57, 401–457 (1953).
D. A. Vladimirov, Boolean Algebras (Nauka, Moscow, 1969) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.I. Chilin and G.B. Levitina, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 63–88.
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134414010040