Let {φs}s∈S be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N . Assume there exists a semigroup {αs}s∈S of weak*-continuous * -endomorphisms of some larger von Neumann algebra M ⊃ N and a projection p ∈ M with N = pM p such that αs(1 − p) ≤ 1 − p for every s ∈ S and φs(y) = pαs(y)p for all y ∈ N . If infs∈S αs(1 − p) = 0 then we show that the map E : M → N defined by E(x) = pxp for x ∈ M induces a complete isometry between the fixed point spaces of {αs}s∈S and {φs}s∈S. Mathematics Subject Classification (2010). Primary 46L55; Secondary 46L05.Let (S, +, 0) be a commutative semigroup with unit 0. Consider the partial preorder on S induced by the semigroup structure as follows. If s, t ∈ S then s ≤ t if and only if there exists r ∈ S such that s + r = t. If X is a Hausdorff topological space and f : S → X is a function, , then lim s∈S f (s) denotes its limit along the directed set (S, ≤), whenever this limit exists.Let M be a von Neumann algebra. Let CP (M ) denote the semigroup of all completely positive, contractive and weak*-continuous linear maps β : M → M . Let also End(M ) be the semigroup of all weak*-continuous *-endomorphisms of M . A family {β s } s∈S ⊂ CP (M ) is called a semigroup if the map s → β s is a unital homomorphism of semigroups from S into CP (M ). Suppose now that {α s } s∈S ⊂ End(M ) is a semigroup. Let p be an orthogonal projection in M such that α s (1 − p) ≤ 1 − p for all s ∈ S. Then one can define, for every s ∈ S, a completely positive mapping on the von Neumann algebra N = pM p as follows: φ s (x) = pα s (x)p ∀x ∈ N. It is clear that {φ s } s∈S ⊂ CP (N ). A short calculation shows that φ s (pxp) = pα s (x)p ∀x ∈ M,