Low-mass stars might offer today the best opportunities to detect and characterize planetary systems, especially those harbouring close-in low-mass temperate planets. Among those stars, TRAPPIST-1 is exceptional since it has seven Earth-sized planets, of which three could sustain liquid water on their surfaces. Here we present new and deep ALMA observations of TRAPPIST-1 to look for an exo-Kuiper belt which can provide clues about the formation and architecture of this system. Our observations at 0.88 mm did not detect dust emission, but can place an upper limit of 23 µJy if the belt is smaller than 4 au, and 0.15 mJy if resolved and 100 au in radius. These limits correspond to low dust masses of ∼10-5 to 10-2 M⊕, which are expected after 8 Gyr of collisional evolution unless the system was born with a >20 M⊕ belt of 100 km-sized planetesimals beyond 40 au or suffered a dynamical instability. This 20 M⊕ mass upper limit is comparable to the combined mass in TRAPPIST-1 planets, thus it is possible that most of the available solid mass in this system was used to form the known planets. A similar analysis of the ALMA data on Proxima Cen leads us to conclude that a belt born with a mass ≳1 M⊕ in 100 km-sized planetesimals could explain its putative outer belt at 30 au. We recommend that future characterizations of debris discs around low-mass stars should focus on nearby and young systems if possible.
