We present ALMA observations of the GQ Lup system, a young Sun-like star with a substellar mass companion in a wide-separation orbit. These observations of 870 $mu$m continuum and CO J=3-2 line emission with beam size $sim0.3$ ($sim45$ AU) resolve the disk of dust and gas surrounding the primary star, GQ Lup A, and provide deep limits on any circumplanetary disk surrounding the companion, GQ Lup b. The circumprimary dust disk is compact with a FWHM of $59pm12$ AU, while the gas has a larger extent with a characteristic radius of $46.5pm1.8$ AU. By forward-modeling the velocity field of the circumprimary disk based on the CO emission, we constrain the mass of GQ Lup A to be $M_* = (1.03pm0.05)*(d/156text{ pc})$ $M_odot$, where $d$ is a known distance, and determine that we view the disk at an inclination angle of $60.5^circpm0.5^circ$ and a position angle of $346^circ pm1^circ$. The $3sigma$ upper limit on the 870 $mu$m flux density of any circumplanetary disk associated with GQ Lup b of $<0.15$ mJy implies an upper limit on the dust disk mass of $<0.04$ $M_oplus$ for standard assumptions about optically thin emission. We discuss proposed mechanisms for the formation of wide-separation substellar companions given the non-detection of circumplanetary disks around GQ Lup b and other similar systems.
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