We report ALMA observations of CO(3-2) emission in the Seyfert 2 double-barred galaxy NGC1433, at the unprecedented spatial resolution of 0.5=24 pc. Our aim is to probe AGN feeding and feedback phenomena through the morphology and dynamics of the gas inside the central kpc. The CO map, which covers the whole nuclear region (nuclear bar and ring), reveals a nuclear gaseous spiral structure, inside the nuclear ring encircling the nuclear stellar bar. This gaseous spiral is well correlated with the dusty spiral seen in Hubble Space Telescope images. The nuclear spiral winds up in a pseudo-ring at 200 pc radius, which might correspond to the inner ILR. Continuum emission is detected at 0.87 mm only at the very centre, and its origin is more likely thermal dust emission than non-thermal emission from the AGN. It might correspond to the molecular torus expected to exist in this Seyfert 2 galaxy. The HCN(4-3) and HCO+(4-3) lines were observed simultaneously, but only upper limits are derived, with a ratio to the CO(3-2) line lower than 1/60 at 3sigma, indicating a relatively low abundance of very dense gas. The kinematics of the gas over the nuclear disk reveal rather regular rotation only slightly perturbed by streaming motions due to the spiral; the primary and secondary bars are too closely aligned with the galaxy major or minor axis to leave a signature in the projected velocities. Near the nucleus, there is an intense high-velocity CO emission feature redshifted to 200 km/s (if located in the plane), with a blue-shifted counterpart, at 2 (100 pc) from the centre. While the CO spectra are quite narrow in the centre, this wide component is interpreted as an outflow involving a molecular mass of 3.6 10^6 Mo and a flow rate 7 Mo/yr. The flow could be in part driven by the central star formation, but is mainly boosted by the AGN through its wind or radio jets.
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