  In this paper, we study the multiplicative behaviour of quantum channels,
mathematically described by trace preserving, completely positive maps on
matrix algebras. It turns out that the multiplicative domain of a unital
quantum channel has a close connection to its spectral properties. A structure
theorem, which reveals the automorphic property of an arbitrary unital quantum
channel on a subalgebra, is presented. Various classes of quantum channels
(irreducible, primitive etc.) are then analysed in terms of this stabilising
subalgebra. The notion of the multiplicative index of a unital quantum channel
is introduced, which measures the number of times a unital channel needs to be
composed with itself for the multiplicative algebra to stabilise. We show that
the maps that have trivial multiplicative domains are dense in completely
bounded norm topology in the set of all unital completely positive maps. Some
applications in quantum information theory are discussed.
