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

Let A be a C∗-algebra generated by a nonself-adjoint idempotent e, and put . 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,M2(C)) of all continuous functions from K to the C∗-algebra M2(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,M2(C)). By replacing C∗-algebra with JB∗-algebra, with the triple spectrum σ(e) of e, and M2(C) with the three-dimensional spin factor C3, similar results are obtained.