_{$*$-Wildness of a Semidirect Product of $\mathcal{F}_2$ and a Finite Group}

Abstract:
To describe how complicated can be the problem of the classification of unitary representations of $*$-algebras, it was introduced by S.A. Kruglyak and Yu.S. Samoilenko a quasi-order $"\succ"$ on $*$-algebras. If $\mathcal{A}\succ\mathcal{B}$ then the classification of unitary representations of $*$-algebra $\mathcal{A}$ contains the problem of unitary classification of representations of $\mathcal{B}$. It was proved, that $C^*(\mathcal{F}_2)$, the enveloping $C^*$-algebra of free group with two generators, majorizes any finitely generated $*$-algebra. Therefore a $*$-algebra $\mathcal{A}$ is called $*$-wild if $\mathcal{A}\succ C^*(\mathcal{F}_2)$. On the hypothesis that every group that has $\mathcal{F}_2$ as a subgroup is $*$-wild we have proved $*$-wildness of a semidirect product $\mathcal{F}_2\rtimes G_f$, where $G_f$ is finite and $\mathcal{F}_2$ is normal subgroups.