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We show that natural noncommutative gauge theory models on$\mathbb{R}^3_\lambda$ can accommodate gauge invariant harmonic terms, thanksto the existence of a relationship between the center of $\mathbb{R}^3_\lambda$and the components of the gauge invariant 1-form canonical connection. Thislatter object shows up naturally within the present noncommutative differentialcalculus. Restricting ourselves to positive actions with covariant coordinatesas field variables, a suitable gauge-fixing leads to a family of matrix modelswith quartic interactions and kinetic operators with compact resolvent. Theirperturbative behavior is then studied. We first compute the 2-point and 4-pointfunctions at the one-loop order within a subfamily of these matrix models forwhich the interactions have a symmetric form. We find that the correspondingcontributions are finite. We then extend this result to arbitrary order. Wefind that the amplitudes of the ribbon diagrams for the models of thissubfamily are finite to all orders in perturbation. This result extends finallyto any of the models of the whole family of matrix models obtained from theabove gauge-fixing. The origin of this result is discussed. Finally, theexistence of a particular model related to integrable hierarchies is indicated,for which the partition function is expressible as a product of ratios ofdeterminants.

Noncommutative gauge theories on $\mathbb{R}^3_\lambda$: Perturbatively finite models