  Stopping-time Banach spaces is a collective term for the class of spaces of
eventually null integrable processes that are defined in terms of the behaviour
of the stopping times with respect to some fixed filtration. From the point of
view of Banach space theory, these spaces in many regards resemble the
classical spaces such as $L^1$ or $C(\Delta)$, but unlike these, they do have
unconditional bases.
  In the present paper we study the canonical bases in the stopping-time spaces
in relation to factorising the identity operator thereon. Since we work
exclusively with the dyadic-tree filtration, this set-up enables us to work
with tree-indexed bases rather than directly with stochastic processes.
\emph{En route} to the factorisation results, we develop general criteria that
allow one to deduce the uniqueness of the maximal ideal in the algebra of
operators on a Banach space. These criteria are applicable to many classical
Banach spaces such as (mixed-norm) $L^p$-spaces, BMO, $\mathrm{SL^\infty}$ and
others.
