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Tensor operators as the irreducible submodules corresponding to the adjoint representation of the quantum algebra Ŭq( su 2) are equipped with q-analogue of the Hilbert–Schmidt scalar product by using the Wigner–Eckart theorem. Then, it is used to show that the adjoint representation of the quantum algebra Ŭq( su 2) is a *-representation.

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Irreducible tensor operators as the irreducible submodules of an adjoint representation of the two-parametric quantum ∗-algebra [Formula: see text] are constructed by using its Jordan–Schwinger formulation on two independent [Formula: see text]-oscillator ∗-algebras. All [Formula: see text]-submodules are equipped with an appropriate Hilbert–Schmidt scalar product with the help of the Wigner–Eckart theorem. We show that with respect to this scalar product, not only the bases of all irreducible submodules of the adjoint representation are orthonormal, but also the adjoint representation is a ∗-representation.