Perfect JC-Algebras

Abstract

Perfect C[Formula: see text]-algebras were introduced by Akeman and Shultz in [Perfect C*-algebras, Mem. Amer. Math. Soc. 55(326) (1985)] and they form a certain subclass of C*-algebras determined by their pure states, and for which the general Stone–Weierstrass conjecture is true. In this paper, we introduce the notion of perfect JC-algebras, and we use the strong relationship between a JC-algebra [Formula: see text] and its universal enveloping C[Formula: see text]-algebra [Formula: see text], to establish that if [Formula: see text] is perfect and [Formula: see text] is of complex type, then [Formula: see text] is perfect. It is also shown that every scattered JC-algebra of complex type is perfect, and the same conclusion holds for every JC-algebra of complex type whose primitive spectrum is Hausdorff.