  We study one-loop perturbative properties of scalar field theories on the
$\rho$-Minkowski space. The corresponding star-product, together with the
involution are characterized from a combination of Weyl quantization and
defining properties of the convolution algebra of the Euclidean group linked to
the coordinate algebra of the $\rho$-Minkowski space. The natural integration
measure linked to the Haar measure of the Euclidean group defines a trace for
the star-product. One-loop properties of the 2-point and 4-point functions for
families of complex-valued scalar field theories on $\rho$-Minkowski space are
examined. For scalar theories with orientable interaction, the 2-point function
is found to receive UV quadratically diverging one-loop corrections in 4
dimensions while no IR singularities generating UV/IR mixing appears. These
however occur in the one-loop corrections to the 4-point function. As well,
one-loop 2-point functions for theories with non-orientable interaction involve
such IR singularities. These results are discussed.
