Derivations with Values in an Ideal \(F\)-Spaces of Measurable Functions

Alimov A. A. , Chilin, V. I.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 1.

Abstract:
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 \(\delta: \mathcal{L}_{\infty}(\Omega, \mu)\to (\mathcal{L}_{0}(\Omega, \mu), \|\cdot\|_{\Omega})\).

Keywords: derivation, an ideal space, \(F\)-norm

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