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

Involutive representations of coordinate algebras and quantum spaces