Conorm and Essential Conorm in C*-Algebras

Abstract

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 $$A/\overline{soc(A)}$$ . It is established that for every von Neumann regular element $$a{\in}A$$ , γ e (a) = max $$\left\{\gamma(a + k) : k \in \overline{soc(A)}\right\}$$ . 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.