  We investigate spectral properties of the tensor products of two quantum
channels defined on matrix algebras. This leads to the important question of
when an arbitrary subalgebra can split into the tensor product of two
subalgebras. We show that for two unital quantum channels $\mathcal{E}_1$ and
$\mathcal{E}_2$ the multiplicative domain of
$\mathcal{E}_1\otimes\mathcal{E}_2$ splits into the tensor product of the
individual multiplicative domains. Consequently, we fully describe the fixed
points and peripheral eigen operators of the tensor product of channels.
Through a structure theorem of maximal unital proper $^*$-subalgebras (MUPSA)
of a matrix algebra we provide a non-trivial upper bound of the 'multiplicative
index' of a unital channel which was recently introduced. This bound gives a
criteria on when a channel cannot be factored into a product of two different
channels. We construct examples of channels which can not be realized as a
tensor product of two channels in any way. With these techniques and results,
we found some applications in quantum error correction.
