The trace of nuclear elements in Banach algebras

Abstract

The classical Ascoli's theorem has proved to be of great interest to many mathematicians and has been the object of many modifications and generalisations. K Vala [14] studied compact and finite elements in a Banach algebra, giving a definition which generalises a theorem in operator theory which states that the mapping: T _,. ATC on the Banach algebra of operators on a Banach space E is compact (of finite rank), if and only if both mappings A and C are compact (finite rank) operators on E. In this paper a different definition for finite (in particular one-dimensional) elements in a Banach algebra, due to J Puhl [10], is given, generalising the following theorems in operator theory: (i) An operator T ¢ 0 on a Banach space if there exists a non-zero functional E is of rank one if and only rT on the Banach algebra of operators on E such that TRT = <rT,R>T for all operators R. (ii) T is of finite rank if and only if it can be written as a finite Slllil of operators of rank one. It is shown that the two different definitions for finite elements, given by Vala and Puhl respectively, coincide. Since most of the results throughout the paper require the Banach algebra to be semi-prime, a condition which is equivalent for this concept is proved. A well-defined trace for one-dimensional elements is introduced provided the Banach algebra is semi-prime. The trace of finite elements is also defined and the results are analogous to those of finite rank operators. Furthermore, the spectrum of a one-dimensional element is shown to consist of exactly two elements and that of a finite element to be finite, by using the same result which is proved to be valid for finite rank operators on a Banach space E. We also prove that if the Banach algebra is semi-prime, the one-dimensional elements and the minimal left (right) ideals are in one to one correspondence. Furthermore, the sole of a semi-prime algebra always exists and equals the class of all finite elements. Nuclear elements are defined in a.natural way and a well-defined nuclear norm is introduced, which dominates the nonn on the Banach algebra. It is shown that if the Banach algebra fulfils certain conditions, the trace can be extended to these elements. However, it is shown that the definition for nuclear elements, given by Vala, implies that of Puhl, but the converse is not necessarily true (even in c*-algebras). The spectrum of a nuclear element is shown to be at most countable, with zero the only point of acctm1ulation.