Narrow Results

Let $𝔍$ and ${𝔍}^{\prime }$ be two ∗-Jordan algebras with identities $I_{𝔍}$ and $I_{{𝔍}^{\prime }}$, respectively, and $e$ a nontrivial ∗-idempotent in $𝔍$. In this paper, we study the characterization of multiplicative ∗-Jordan-type maps. In particular, we provide a characterization in the case of unital prime associative algebra endowed with an involution.

Perfect C^∗-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 A and its universal enveloping C^∗-algebra $C^{\ast }(A)$, to establish that if $C^{\ast }(A)$ is perfect and A is of complex type, then A 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.