Where to find the image of a derivation

Abstract

Abstract. With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the noncommutative Singer-Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image.

Elementary operators - still not elementary?

“…The last-quoted paper [28] by Pták is one of many articles by a variety of authors in which the origins of spectrally bounded operators can be found. The concept itself was introduced in [18] with the goal to unify various ideas and to progress towards a more systematic treatment which was then started in [22]. Pták's aim was ".…”

Spectrally bounded generalized inner derivations

“…A spectrally infinitesimal derivation on A maps into the radical. This can be found in various places (see, e.g., [3]). …”

“…This concept arises in connection with the non-commutative Singer-Wermer conjecture stating that the image ÔA of any derivation 6 on A is contained in the Jacobson radical rad(^) if and only if xôx -(ôx)x £ rad(A) for all x £ A . Although some progress on this conjecture has been made lately (see [6] and the references in [3]), it is still open and seems hard to verify.…”

“…In [3], a linear mapping T on a complex unital Banach algebra A was defined to be spectrally bounded if r(Tx) < Mr(x) for all x £ A and some M > 0, where r(-) denotes the spectral radius. This concept arises in connection with the non-commutative Singer-Wermer conjecture stating that the image ÔA of any derivation 6 on A is contained in the Jacobson radical rad(^) if and only if xôx -(ôx)x £ rad(A) for all x £ A .…”

On generalized derivations in Banach algebras

“…For details, we refer the readers to [17,18], and the references therein. We also mention the work of Curto and Mathieu [11], where spectrally bounded generalized inner derivations were investigated.…”