Abstract

We describe when a general complex algebraic ${}^{\ast }$-algebra is pre-$C^{\ast }$-normed, and investigate its structure when the ${}^{\ast }$-algebra is a Baer ${}^{\ast }$-ring in the presence of algebraicity. Our main result is that every complex algebraic Baer ${}^{\ast }$-algebra can be decomposed as a direct sum $M\oplus B$, where $M$ is a finite-dimensional Baer ${}^{\ast }$-algebra and $B$ is a commutative algebraic Baer ${}^{\ast }$-algebra. The summand $M$ is ${}^{\ast }$-isomorphic to a finite direct sum of full complex matrix algebras of size at least $2\times 2$. The commutative summand $B$ is ${}^{\ast }$-isomorphic to the linear span of the characteristic functions of the clopen sets in a Stonean topological space.

As an application we show that a group $G$ is finite exactly when the complex group algebra $ℂ[G]$ is an algebraic Baer ${}^{\ast }$-algebra.

Keywords

Mathematical Subject Classification

Milestones

Authors