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Derivations mapping into the socle, III

100%

Boudi N. , Šemrl P.

nr 2

141-155

Let A be a Banach algebra, and let d: A → A be a continuous derivation such that each element in the range of d has a finite spectrum. In a series of papers it has been proved that such a derivation is an inner derivation implemented by an element from the socle modulo the radical of A (a precise formulation of this statement can be found in the Introduction). The aim of this paper is twofold: we extend this result to the case where d is not necessarily continuous, and we give a complete description of such maps in the semisimple case.

Finite rank elements in semisimple Banach algebras

100%

Brešar M. , Šemrl P.

nr 3

287-298

Let A be a semisimple Banach algebra. We define the rank of a nonzero element a in the socle of A to be the minimum of the number of minimal left ideals whose sum contains a. Several characterizations of rank are proved.

On operator bands

61%

Drnovšek R. , Livshits L. , MacDonald G. W. , Mathes B. , Radjavi H. , Šemrl P.

2000

tom 139

nr 1

91-100

A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.

On Jordan *-derivations and an application

50%

Šemrl P.

tom 59

nr 2

241-251