  We investigate simpliciality of function spaces without constants. We prove,
in particular, that several properties characterizing simpliciality in the
classical case differ in this new setting. We also show that it may happen that
a given point is not represented by any measure pseudosupported by the Choquet
boundary, illustrating so limits of possible generalizations of the
representation theorem. Moreover, we address the abstract Dirichlet problem in
the new setting and establish some common points and nontrivial differences
with the classical case.
