C∗- and JB∗-algebras generated by a nonself-adjoint idempotent

Abstract

Let A be a C * -algebra generated by a nonself-adjoint idempotent e, and put K := sp( √ e * e) \ {0}. It is known that K is a compact subset of [1, ∞[ whose maximum element is greater than 1, and that, in general, no more can be said about K. We prove that, if 1 does not belong to K, then A is * -isomorphic to the C * -algebra C(K, M 2 (C)) of all continuous functions from K to the C * -algebra M 2 (C) (of all 2 × 2 complex matrices), and that, if 1 belongs to K, then A is * -isomorphic to a distinguished proper C * -subalgebra of C(K, M 2 (C)). By replacing C * -algebra with J B * -algebra, sp( √ e * e) \ {0} with the triple spectrum σ (e) of e, and M 2 (C) with the three-dimensional spin factor C 3 , similar results are obtained.