Derivations with values in an ideal $F$-spaces of measurable functions

A. A. Alimov   ; V. I. Chilin

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 21-29

Résumé

It is known that any derivation on a commutative von Neumann algebra $ \mathcal {L}_{\infty} (\Omega, \mu)$ is identically equal to zero. At the same time, the commutative algebra $\mathcal {L}_{0}(\Omega, \mu)$ of complex measurable functions defined on a non-atomic measure space $(\Omega,\mu)$ admits non-zero derivations. Besides, every derivation on $\mathcal{L}_{\infty}(\Omega, \mu)$ with the values in an ideal normed subspace $X \subset \mathcal{L}_{0}(\Omega,\mu)$ is equal to zero. The same remains true for an ideal quasi-normed subspace $X \subset\mathcal{L}_{0}(\Omega, \mu)$. Naturally, there is the problem of describing the class of ideal $F$-normed spaces $X \subset \mathcal{L}_{0}(\Omega, \mu)$ for which there is a non-zero derivation on $\mathcal{L}_{\infty}(\Omega, \mu)$ with the values in $X $. We give necessary and sufficient conditions for a complete ideal $F$-normed spaces $X$ to be such that there is a non-zero derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu) \to X$. In particular, it is shown that if the $F$-norm on $X$ is order semicontinuous, each derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu) \to X$ is equal to zero. At the same time, existence of a non-atomic idempotent $0\neq e \in X$, $\mu(e) < \infty$ for which the measure topology in $e \cdot X$ coincides with the topology generated by the $F$-norm implies the existence of a non-zero derivation $\delta: \mathcal{L}_{\infty}(\Omega, \mu)\to X$. Examples of such ideal $F$-normed spaces are algebras $\mathcal{L}_{0}(\Omega, \mu)$ with non-atomic measure spaces $(\Omega, \mu)$ equipped with the $F$-norm $\| f\|_{\Omega} = \int_{\Omega} \frac {| f |} {1+ | f |} d\mu $. For such ideal $ F$-spaces there is at least a continuum of pairwise distinct non-zero derivations $\delta: \mathcal{L}_{\infty}(\Omega, \mu)\to (\mathcal{L}_{0}(\Omega, \mu), \|\cdot\|_{\Omega})$.

Bibliographie

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

[2] Ber A. F., “Derivations on commutative regular algebras”, Sib. Adv. Math., 21:3 (2011), 161–169 | DOI | MR

[3] 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

[4] Bennet C., Sharpley R., Interpolation of Operators, Acad. Press Inc., N. Y., 1988 | MR | Zbl

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

[6] Dykema K., Sukochev F., Zanin D., Algebras of log-integrable functions and operators, 10 Sep. 2015, 11 pp., arXiv: 1509.03360v1 [math.OA] | MR

[7] Kalton N. J., Peck N. T., Roberts James W., An $F$-space sampler, London Math. Soc. Lect. Note Ser., 89, Cambridge Univ. Press, Cambridge, 1984 | MR | Zbl

[8] Kusraev A. G., Dominated Operators, Math. and its Appl., 519, Kluwer Academic Publishers, Dordrecht, 2000 | MR | MR | Zbl

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

[10] Rudin W., Functional Analysis, McGraw-Hill Book Company, New York, 1973 | MR | MR | Zbl

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

[12] Vladimirov D. A., Boolean Algebras in Analysis, Math. Appl., 540, Springer, Dordrecht, 2002, 604 pp. | MR

[13] Vulikh B. Z., Introduction to the theory of partially ordered spaces, Wolters-Noordhoff Sci. Publ. Ltd., Groningen, 1967, 387 pp. | MR | Zbl