On the complexity of description of representations of $*$-algebras generated by idempotents

In this paper, we introduce a quasiorder $\succ$ (majorization) on $*$-algebras with respect to the complexity of description of their representations. We show that $C^*({\mathcal F}_2) \succ \mathfrak A$ for any finitely generated $*$-algebra $\mathfrak A$ (algebras $\mathfrak B$ such that $\mathfrak B\succ C^*({\mathcal F}_2)$ are called $*$-wild). We show that the $*$-algebra generated by orthogonal projections $p$, $p_1$, $p_2$, ..., $p_n$ ($p_ip_j=0$ for $i\neq j$) is $*$-wild if $n\geq 2$. We also prove that $*$-algebras generated by a pair of idempotents and an orthogonal projection, or by a pair of idempotents $q_1$, $q_2$ ( $q_1q_2=q_2 q_1=0$), etc., are $*$-wild.