R. Brits

Uniqueness and spectral variation in Banach algebras

Abstract Let A be a complex semisimple Banach algebra with identity. We explore the situation whereby a portion of the elements of A have the same spectrum under multiplication by a ∈ A, as under multiplication by b ∈ A; and when this situation implies that a and b are the same. In particular we show that if the spectrum of ax equals the spectrum of bx for all x with a spectral radius away from the identity less than 1, then a and b coincide. By way of examples we show that this is the best situation possible in general. In another result we show that in the case where these spectra are finite, the assumption need only hold for an arbitrarily small open set in A, with the same conclusion. Additive versions of these results are also discussed.

On the Multiplicative Spectral Characterization of the Jacobson Radical

For a unital Banach algebra A over C, we give improvements on the well-known multiplicative spectral characterization of the Jacobson radical, (1), in terms of several spectral parameters. In particular, we show that every non-invertible element a ∈ A for which the number of elements in the spectrum of ax is less or equal to the number of elements in the spectrum of x, for all x in an arbitrary small neighborhood of the unit, must belong to the Jacobson radical.

The Shoda-completion of a Banach algebra

Abstract In stark contrast to the case of finite rank operators on a Banach space, the socle of a general, complex, semisimple and unital Banach algebra A may exhibit the ‘pathological’ property that not all traceless elements of the socle of A can be expressed as the commutator of two elements belonging to the socle. The aim of this paper is to show how one may develop an extension of A which removes the aforementioned problem. A naive way of achieving this is to simply embed A in the algebra of bounded linear operators on A, i.e. the natural embedding of A into . But this extension is so large that it may not preserve the socle of A in the extended algebra . Our proposed extension, which we shall call the Shoda-completion of A, is natural in the sense that it is small enough for the socle of A to retain the status of socle elements in the extension.

Applications of the growth characteristics induced by the spectral distance

Abstract Let A be a complex unital Banach algebra. Using a connection between the spectral distance and the growth characteristics of a certain entire map into A, we derive a generalization of Gelfand’s famous power boundedness theorem. Elaborating on these ideas, with the help of a Phragm´en-Lindel¨of device for subharmonic functions, it is then shown, as the main result, that two normal elements of a C∗-algebra are equal if and only if they are quasinilpotent equivalent.