How does the Kraus operator act on a state given by the direct sum of different states?

Let’s say i have a state of the form
rho_net = q *({summation over i} pi |i><i|)⊕ (1-q)rho
here i = 1-n and the so called "basis" in which the density matrix rho is defined is of dimension d-n , where d is my total dimension of the Hilbert space and d-n is only a part.
sum over pi = 1
.
my main question is will the kraus operators act on the net rho as a whole or can we make them operate individually on the two different states whose direct sum has been taken?