  We introduce a new decomposition of quantum channels acting on group
algebras, which we term Kraus-like (operator) decompositions. We motivate this
decomposition with a general nonexistence result for Kraus operator
decompositions in this setting. Given a length function which is a class
function on a finite group, we construct a corresponding Kraus-like
decomposition. We prove that this Kraus-like decomposition is \textit{convex}
(meaning its coefficients are nonnegative and satisfy a sum rule) if and only
if the length is conditionally negative definite. For a general finite group,
we prove a stability condition which shows that the existence of a convex
Kraus-like decomposition for all $t>0$ small enough necessarily implies
existence for all time $t>0$. Using the stability condition, we show that for a
general finite group, conditional negativity of the length function is
equivalent to a set of semidefinite linear constraints on the length function.
Our result implies that in the group algebra setting, a semigroup $P_t$ induced
by a length function which is a class function is a quantum channel for all
$t\geq 0$ if and only if it possesses a convex Kraus-like decomposition for all
$t>0$.
