  Low mass stars might offer today the best opportunities to detect and
characterise 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$\mu$Jy 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 $\sim10^{-5}-10^{-2}$ $M_\oplus$, which are expected after 8
Gyr of collisional evolution unless the system was born with a $>20$ $M_\oplus$
belt of 100 km-sized planetesimals beyond 40 au or suffered a dynamical
instability. This $20$ $M_\oplus$ 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 $\gtrsim1$ $M_\oplus$ in 100 km-sized planetesimals could
explain its putative outer belt at 30 au. We recommend that future
characterisations of debris discs around low mass stars should focus on nearby
and young systems if possible.
