On $\ast$-representations of the $\mathbf Z_2$-graded extension of the quantum group $U_q(2)$

The possibility of introducing an involution for the $Z_2$-graded extension of the function algebra on the quantum group $GL_q(N)$ is discussed. The involution permits the quantum group $GL_q(N)$ to have the compact form which is $U_q(N)$. However, the compact form related to $SU_q(N)$ is not allowed. $\ast$-representations of the $Z_2$-graded extension of $U_q(2)$ in a Hilbert space are constructed. The operators corresponding to the differentials are expressed as derivations on the space of all irreducible $\ast$-representations of $U_q(2)$.