Derivations on Banach $*$-ideals in von Neumann algebras

A. F. Ber ; V. I. Chilin ; F. A. Sukochev

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 23-28

Résumé

It is known that any derivation $\delta: \mathcal M \to \mathcal M$ on the von Neumann algebra $\mathcal M$ is an inner, i. e. $\delta(x) := \delta_a(x) =[a, x] =ax -xa$, $x \in \mathcal M$, for some $a \in \mathcal M$. If $H$ is a separable infinite-dimensional complex Hilbert space and $\mathcal K(H)$ is a $C^*$-subalgebra of compact operators in $C^*$-algebra $\mathcal B(H)$ of all bounded linear operators acting in $H$, then any derivation $\delta: \mathcal K(H) \to \mathcal K(H)$ is a spatial derivation, i.e. there exists an operator $ a \in \mathcal B(H)$ such that $\delta(x) = [x, a]$ for all $x \in K(H)$. In addition, it has recently been established by Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. (JMAA, 2013) that any derivation $\delta: \mathcal{E}\to \mathcal{E}$ on Banach symmetric ideal of compact operators $\mathcal{E} \subseteq \mathcal K(H)$ is a spatial derivation. We show that the same result is also true for an arbitrary Banach $*$-ideal in every von Neumann algebra $\mathcal{M}$. More precisely: If $\mathcal{M}$ is an arbitrary von Neumann algebra, $\mathcal{E}$ be a Banach $*$-ideal in $\mathcal{M}$ and $\delta\colon \mathcal{E}\to \mathcal{E}$ is a derivation on $\mathcal{E}$, then there exists an element $ a \in \mathcal{M}$ such that $\delta(x) = [x, a]$ for all $x \in \mathcal{E}$, i. e. $\delta $ is a spatial derivation.

Bibliographie

[1] Ber A. F., Chilin V. I., Levitina G. B., “Derivations with values in quasi-normed bimodules of locally measurable operators”, Sib. Adv. Math., 25:3 (2015), 169–178 | DOI | MR

[2] Ber A. F., Chilin V. I., Levitina G. B., Sukochev F. A., “Derivations with values in quasi-normed bimodules of locally measurable operators”, J. Math. Anal. Appl., 397:2 (2013), 628–643 | DOI | MR | Zbl

[3] Bratelli O., Robinson D. W., Operator algebras and quantum statistical mechanics 1, Springer-Verlag, N. Y., 1979 | MR

[4] Chilin V. I., Levitina G. B., “Derivations on ideals in commutative $AW^*$-algebras”, Sib. Adv. Math., 24:1 (2014), 26–42 | DOI | MR

[5] Gohberg I., Krein M., Introduction to the theory of linear nonselfadjoint operators, Translat. of Math. Monogr., 18, Amer. Math. Soc., Providence, R.I., 1969 | DOI | MR | Zbl

[6] Kalton N., Sukochev F., “Symmetric norms and spaces of operators”, J. Reine Angew. Math., 621 (2008), 81–121 | MR | Zbl

[7] Kusraev A. G., “Automorphisms and derivations on a universally complete complex $f$-algebra”, Sib. Math. J., 47:1 (2006), 77–85 | DOI | MR | Zbl

[8] Sakai S., $C^*$-Algebras and $W^*$-Algebras, Springer-Verlag, N. Y.–Heidelberg–Berlin, 1971 | MR | Zbl

[9] Schatten R., Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, 27, Second printing, Springer-Verlag, Berlin–N. Y., 1970, 98 pp. | MR | Zbl

[10] Simon B., Trace ideals and their applications, Math. Surveys and Monogr., 120, Second edition, Amer. Math. Soc., Providence, R. I., 2005 | MR | Zbl

[11] Strătilă S., Zsido L., Lectures on von Neumann algebras, Translated from the Romanian by Silviu Teleman, Revision of the 1975 original, Editura Academiei, Bucharest, 1979 | MR | Zbl

[12] Zsido L., “The norm of a derivation in a $W^*$-algebra”, Proc. Amer. Math. Soc., 38 (1973), 147–150 | DOI | MR | Zbl

[13] Takesaki M., Theory of Operator Algebras I, Springer-Verlag, N. Y.–Heidelberg–Berlin, 1979 | MR | Zbl