  We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to
quantum spaces with $\frak{su}(2)$ noncommutativity can be conveniently
represented by $SO(3)$-covariant poly-differential involutive representations.
We show that the quantized plane waves obtained from the quantization map
action on the usual exponential functions are determined by polar decomposition
of operators combined with constraint stemming from the Wigner theorem for
$SU(2)$. Selecting a subfamily of $^*$-representations, we show that the
resulting star-product is equivalent to the Kontsevich product for the Poisson
manifold dual to the finite dimensional Lie algebra $\mathfrak{su}(2)$. We
discuss the results, indicating a way to extend the construction to any
semi-simple non simply connected Lie group and present noncommutative scalar
field theories which are free from perturbative UV/IR mixing.
