The structure of algebraic Baer $^*$-algebras

Zsolt Szűcs and Balázs Takács

Vol. 316 (2022), No. 2, 431–452

DOI: 10.2140/pjm.2022.316.431

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

algebraic algebra, von Neumann regular algebra, Baer *-algebra

Mathematical Subject Classification

Primary: 16W10

Secondary: 22D15, 46L99

Milestones

Received: 13 August 2021

Revised: 8 November 2021

Accepted: 21 December 2021

Published: 6 April 2022