#### Vol. 316, No. 2, 2022

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The structure of algebraic Baer $^*$-algebras

### Zsolt Szűcs and Balázs Takács

Vol. 316 (2022), No. 2, 431–452
##### 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 $2×2$. 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 $ℂ\left[G\right]$ is an algebraic Baer ${}^{\ast }$-algebra.

##### Keywords
algebraic algebra, von Neumann regular algebra, Baer *-algebra
##### Mathematical Subject Classification
Primary: 16W10
Secondary: 22D15, 46L99