Commutators with finite spectrum
The purpose of this paper is to study derivations d, d′ defined on a Banach algebra A such that the spectrum σ([dx,d′x]) is finite for all x ∈ A. In particular we show that if the algebra is semisimple, then there exists an element a in the socle of A such that [d,d′] is the inner derivation implemented by a.
Identifying Derivations Through the Spectra of Their Values
We consider the relationship between derivations d and g of a Banach algebra B that satisfy $${{\sigma}(g(x)) \subseteq {\sigma}(d(x))}$$ for every $${x\in B}$$ , where σ( . ) stands for the spectrum. It turns out that in some basic situations, say if B = B(X), the only possibilities are that g = d, g = 0, and, if d is an inner derivation implemented by an algebraic element of degree 2, also g = −d. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general C*-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition $${\|[b,x]\|\leq M\|[a,x]\|}$$ for all selfadjoint elements x from a C*-algebra B, where $${a,b\in B}$$ and a is normal.
Derivations with Invertible Values
In this paper we study a question which, although somewhat special, has the virtue that its answer can be given in a very precise, definitive, and succinct way. It shows that the structure of a ring is very tightly determined by the imposition of a special behavior on one of its derivations. The problem which we shall examine is: Suppose that R is a ring with unit element, 1, and that d ≠ 0 is a derivation of R such that for every x ∊ R, d(x) = 0 or d(x) is invertible in R; must R then have a very special structure? As we shall see, the answer to this question is yes, in particular we show that except for a special case which occurs when 2R = 0, R must be a division ring D or the ring D 2 of 2 × 2 matrices over a division ring.