  We present results of dust continuum and [CII]$\,158\,{\rm \mu m}$ emission
line observations of a remarkably UV-luminous ($M_{\rm UV}=-21.6$) galaxy at
$z=10.603$: GN-z11. Using the Northern Extended Millimeter Array (NOEMA),
observations have been carried out over multiple observing cycles. We achieved
a high sensitivity resulting in a $\lambda_{\rm rest}=160\,{\rm \mu m}$
continuum $1\,\sigma$ depth of $13.0\,\rm{\mu Jy/beam}$ and a [CII] emission
line $1\,\sigma$ sensitivity of $31\,\rm{mJy/beam\,km/s}$ using $50\,\rm{km/s}$
binning with a $\sim 2\,{\rm arcsec}$ synthesized beam. Neither dust continuum
nor [CII]$\,158\,{\rm \mu m}$ line emission are detected at the expected
frequency of $\nu_{\rm [CII]} = 163.791\,\rm{GHz}$ and the sky location of
GN-z11. The upper limits show that GN-z11 is neither luminous in $L_{\rm IR}$
nor $L_{\rm [CII]}$, with a dust mass $3\,\sigma$ limit of ${\rm log}(M_{\rm
dust}/{\rm M_{\odot}}) < 6.5-6.9$ and with a [CII] based molecular gas mass
$3\,\sigma$ limit of ${\rm log}(M_{\rm mol,[CII]}/{\rm M_{\odot}}) < 9.3$.
Together with radiative transfer calculations, we also investigated the
possible cause of the dust poor nature of the GN-z11 showed by the blue color
in the UV continuum of GN-z11 ($\beta_{\rm UV}=-2.4$), and found that
$\gtrsim3\times$ deeper observations are crucial to study dust production at
very high-redshift. Nevertheless, our observations show the crucial role of
deep mm/submm observations of very high redshift galaxies to constrain multiple
phases in the interstellar medium.
