JB-algebras of rank zero

Abstract

A unital JB-algebra A is defined to be of rank zero if the set of invertible elements is dense in A. A non-unital JB-algebra A is said to be of rank zero if its unitization A ⊕R1 is so. We show that a unital JB-algebra A is of rank zero if and only if the set of elements with finite spectrum is dense in A if and only if every inner ideal of A admits an approximate identity (not necessarily increasing) consisting of projections. Moreover, we establish that zero rank is inherited by every closed ideal and every quotient algebra.