Extensions of Inductive Limits of Circle Algebras

A classic result of L. G. Brown [ 3 ] and G. Elliott [ 7 ] says that every extension of two AF‐algebras is again an AF‐algebra. We generalize this result to the larger class of C∗‐algebras which are inductive limits of circle algebras and have real rank zero. Let E be an extension of C∗‐algebras A and B ,0 → A → E → B → 0 ,where A and B have real rank zero and are inductive limits of circle algebras. If E has real rank zero and stable rank one, then it is an inductive limit of circle algebras. Moreover, E has real rank zero and is an inductive limit of circle algebras if and only if the extension satisfies the condition that the index maps K j ( B ) → K 1− j ( A ) for j = 0,1, are zero.