Compactness of derivations from commutative Banach algebras

We consider the compactness of derivations from commutative Banach algebras into their dual modules. We show that if there are no compact derivations from a commutative Banach algebra, $A$, into its dual module, then there are no compact derivations from $A$ into any symmetric $A$-bimodule; we also prove analogous results for weakly compact derivations and for bounded derivations of finite rank. We then characterise the compact derivations from the convolution algebra $\ell^1(\Z_+)$ to its dual. Finally, we give an example (due to J. F. Feinstein) of a non-compact, bounded derivation from a uniform algebra $A$ into a symmetric $A$-bimodule

Hasse--Schmidt derivations versus classical derivations

In this paper we survey the notion and basic results on multivariate Hasse--Schmidt derivations over arbitrary commutative algebras and we associate to such an object a family of classical derivations. We study the behavior of these derivations under the action of substitution maps and we prove that, in characteristic $0$, the original multivariate Hasse--Schmidt derivation can be recovered from the associated family of classical derivations. Our constructions generalize a previous one by M. Mirzavaziri in the case of a base field of characteristic $0$.Comment: Dedicated to L\^e D\~ung Tr\'ang; final version; 2 references added; minor corrections. arXiv admin note: text overlap with arXiv:1807.10193, arXiv:1903.0898

On derivations with respect to finite sets of smooth functions

The purpose of this paper is to show that functions that derivate the two-variable product function and one of the exponential, trigonometric or hyperbolic functions are also standard derivations. The more general problem considered is to describe finite sets of differentiable functions such that derivations with respect to this set are automatically standard derivations