Department of Cybernetics,

Kyiv Taras Shevchenko National University,

64 Volodymyrska Str.,

01017 Kyiv,

UKRAINE

E-mail: jushenko@mail.ru

_{$*$-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.