Geodesic
Derivations on ideals in commutative $AW^*$-algebras
Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 63-88.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MT_2013_16_1_a4,
     author = {G. B. Levitina and V. I. Chilin},
     title = {Derivations on ideals in commutative $AW^*$-algebras},
     journal = {Matemati\v{c}eskie trudy},
     pages = {63--88},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2013_16_1_a4/}
}
TY  - JOUR
AU  - G. B. Levitina
AU  - V. I. Chilin
TI  - Derivations on ideals in commutative $AW^*$-algebras
JO  - Matematičeskie trudy
PY  - 2013
SP  - 63
EP  - 88
VL  - 16
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2013_16_1_a4/
LA  - ru
ID  - MT_2013_16_1_a4
ER  - 
%0 Journal Article
%A G. B. Levitina
%A V. I. Chilin
%T Derivations on ideals in commutative $AW^*$-algebras
%J Matematičeskie trudy
%D 2013
%P 63-88
%V 16
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2013_16_1_a4/
%G ru
%F MT_2013_16_1_a4
G. B. Levitina; V. I. Chilin. Derivations on ideals in commutative $AW^*$-algebras. Matematičeskie trudy, Tome 16 (2013) no. 1, pp. 63-88. http://geodesic.mathdoc.fr/item/MT_2013_16_1_a4/

[1] Ber A. F., Sukochev F. A., Chilin V. I., “Differentsirovaniya v kommutativnykh regulyarnykh algebrakh”, Matem. zametki, 75:3 (2004), 453–457 | DOI | MR | Zbl

[2] Vladimirov D. A., Bulevy algebry, Nauka, M., 1969 | MR

[3] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984 | MR | Zbl

[4] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978 | MR

[5] Kusraev A. G., Vektornaya dvoistvennost i ee prilozheniya, Nauka, Novosibirsk, 1985 | MR | Zbl

[6] Kusraev A. G., Mazhoriruemye operatory, Nauka, M., 2003 | MR

[7] Kusraev A. G., “Avtomorfizmy i differentsirovaniya v rasshirennoi kompleksnoi $f$-algebre”, Sib. matem. zhurn., 47:1 (2006), 97–107 | MR | Zbl

[8] Chilin V. I., “Chastichno uporyadochennye berovskie involyutivnye algebry”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat. Nov. dostizh., 27, VINITI, M., 1985, 99–128 | MR | Zbl

[9] Albeverio S., Ayupov Sh., Kudaybergenov K. K., “Structure of derivations on various algebras of measurable operators for type I von Neumann algebras”, J. Funct. Anal., 256:9 (2009), 2917–2943 | DOI | MR | Zbl

[10] Ber A. F., Chilin V. I., Sukochev F. A., “Nontrivial derivations on commutative regular algebras”, Extracta Math., 21:2 (2006), 107–147 | MR | Zbl

[11] Ber A. F., Sukochev F. A., Derivations in the Banach ideals of $\tau$-compact operators, 2012, arXiv: 1204.4052v1[math.OA]

[12] Berberian S. K., Baer $*$-Rings, Die Grundlehren der Mathematischen Wissenschaften, 195, Springer-Verlag, New York–Berlin, 1972 | DOI | MR | Zbl

[13] Gutman A. E., Kusraev A. G., Kutateladze S. S., “The Wickstead problem”, Sib. Èlectron. Mat. Izv., 5 (2008), 293–333 | MR

[14] Olesen D., “Derivations of $AW^*$-algebras are inner”, Pacific J. Math., 53:1 (1974), 555–561 | DOI | MR | Zbl

[15] Sakai S., $C^*$-Algebras and $W^*$-Algebras, Springer-Verlag, New York, 1971 | MR | Zbl

[16] Segal I. E., “A noncommutative extension of abstract integration”, Ann. of Math., 57:2 (1953), 401–457 | DOI | MR | Zbl