The structure of algebraic Baer ∗-algebras

The purpose of this note is to describe when a general complex algebraic $^*$-algebra pre-$C^*$-normed, and investigate their structure the $^*$-algebras are Baer $^*$-rings in addition algebraicity. As main result we prove following theorem for $^*$-algebras: every kind can be decomposed as direct sum $M\oplus B$, where $M$ finite dimensional $B$ commutative $^*$-algebra. summand $^*$-isomorphic full matrix algebras size at least $2\times2$. linear span characteristic functions clopen sets Stonean topological space. an application show that group $G$ exactly algebra $\mathbb{C}[G]$