  This work is motivated by Radulescu's result on the comparison of C*-tensor
norms on C*(F_n) x C*(F_n). For unital C*-algebras A and B, there are natural
inclusions of A and B into their unital free product, their maximal tensor
product and their minimal tensor product. These inclusions define three
operator system structures on the internal sum A+B, the first of which we
identify as the coproduct of A and B in the category of operator systems.
Partly using ideas from quantum entanglement theory, we prove various
interrelations between these three operator systems. As an application, the
present results yield a significant improvement over Radulescu's bound on
C*(F_n) x C*(F_n). At the same time, this tight comparison is so general that
it cannot be regarded as evidence for a positive answer to the QWEP conjecture.
