FAUST VIII. The protostellar disk of VLA 1623-2417 W and its streamers imaged by ALMA

More than 50% of solar-mass stars form in multiple systems. It is therefore crucial to investigate how multiplicity affects the star and planet formation processes at the protostellar stage. We report continuum and C$^{18}$O (2-1) observations of the VLA 1623-2417 protostellar system at 50 au angular resolution as part of the ALMA Large Program FAUST. The 1.3 mm continuum probes the disks of VLA 1623A, B, and W, and the circumbinary disk of the A1+A2 binary. The C$^{18}$O emission reveals, for the first time, the gas in the disk-envelope of VLA 1623W. We estimate the dynamical mass of VLA 1623W, $M_{\rm dyn}=0.45\pm0.08$ M$_{\odot}$, and the mass of its disk, $M_{\rm disk}\sim6\times10^{-3}$ M$_{\odot}$. C$^{18}$O also reveals streamers that extend up to 1000 au, spatially and kinematically connecting the envelope and outflow cavities of the A1+A2+B system with the disk of VLA 1623W. The presence of the streamers, as well as the spatial ($\sim$1300 au) and velocity ($\sim$2.2 km/s) offset of VLA 1623W suggest that either sources W and A+B formed in different cores, interacting between them, or that source W has been ejected from the VLA 1623 multiple system during its formation. In the latter case, the streamers may funnel material from the envelope and cavities of VLA 1623AB onto VLA 1623W, thus concurring to set its final mass and chemical content.