  Gluings of completely positive maps (CPMs) are defined and investigated. As a
brief description of this concept consider a pair of `evolution machines', each
with the ability to evolve the internal state of a `particle' inserted into its
input. Each of these machines is characterized by a channel describing the
operation the internal state has experienced when the particle is returned at
the output. Suppose a particle is put in a superposition between the input of
the first and the second machine. Here it is shown that the total evolution
caused by a pair of such devices is not uniquely determined by the channels of
the two machines. Such `global' channels describing the machine pair are
examples of gluings of the two single machine channels. Under the limiting
assumption that all involved Hilbert spaces are finite-dimensional, an
expression which generates all subspace preserving gluings of a given pair of
CPMs, is derived. The nature of the non-uniqueness of gluings and its relation
to a proposed definition of subspace locality, is discussed.
