Conorm and Essential Conorm in C*-Algebras

Let A be a unital C^*-algebra with non-zero socle (soc(A)). We introduce the essential conorm of an element a in A (denoted by γ _e (a)), as the conorm of the element π(a), where π denotes the canonical projection of A onto . It is established that for every von Neumann regular element , γ _e (a) = max . We characterize the continuity points of the conorm and essential conorm for extremally rich C^*-algebras. Some formulae for the distance from zero to the generalized spectrum and Atkinson spectrum are also obtained. Authors partially supported by I+D MEC projects no. MTM2005-02541 and MTM2007-65959, and Junta de Andalucía grants FQM0199 and FQM1215.