  Non-global logarithms arise from the sensitivity of collider observables to
soft radiation in limited angular regions of phase space. Their resummation to
next-to-leading logarithmic (NLL) order has been a long standing problem and
its solution is relevant in the context of precision all-order calculations in
a wide variety of collider processes and observables. In this article, we
consider observables sensitive only to soft radiation, characterised by the
absence of Sudakov double logarithms, and we derive a set of
integro-differential equations that describes the resummation of NLL soft
corrections in the planar, large-$N_c$ limit. The resulting set of evolution
equations is derived in dimensional regularisation and we additionally provide
a formulation that is manifestly finite in four space-time dimensions. The
latter is suitable for a numerical integration and can be generalised to treat
other infrared-safe observables sensitive solely to soft wide-angle radiation.
We use the developed formalism to carry out a fixed-order calculation to ${\cal
O}(\alpha_s^2)$ in full colour for both the transverse energy and energy
distribution in the interjet region between two cone jets in $e^+e^-$
collisions. We find that the expansion of the resummed cross section correctly
reproduces the logarithmic structure of the full QCD result.
