We present ALMA observations of the 870$mu$m continuum and CO(4-3) line emission in the core of the galaxy cluster ClJ1449+0856 at z=2, a NIR-selected, X-ray detected system in the mass range of typical progenitors of todays massive clusters. The 870$mu$m map reveals six F$_{870mu m}$ > 0.5 mJy sources spread over an area of 0.07 arcmin$^2$, giving an overdensity of a factor ~10 (6) with respect to blank field counts down to F$_{870mu m}$ > 1 (0.5) mJy. On the other hand, deep CO(4-3) follow-up confirms membership of three of these sources, but suggests that the remaining three, including the brightest 870$mu$m sources in the field (F$_{870mu m}gtrsim$2 mJy), are likely interlopers. The measurement of 870$mu$m continuum and CO(4-3) line fluxes at the positions of previously-known cluster members provides a deep probe of dusty star formation occurring in the core of this high-redshift structure, adding up to a total SFR~700$pm$100 M$_{odot}$/yr and yielding an integrated star formation rate density of ~10$^4$ M$_{odot}$/yr/Mpc$^3$, five orders of magnitude larger than in the field at the same epoch, due to the concentration of star-forming galaxies in the small volume of the dense cluster core. The combination of these observations with previously available HST imaging highlights the presence in this same volume of a population of galaxies with already suppressed star formation. This diverse composition of galaxy populations in ClJ1449+0856 is especially highlighted at the very cluster center, where a complex assembly of quiescent and star-forming sources is likely forming the future Brightest Cluster Galaxy.
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