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We consider functions on the unit disc whose images are contained in a strip of the complex plane. Under an additional condition, such functions are constants. We also consider appropriate operator valued versions. Applications are found to the theory of semigroups. [ABSTRACT FROM AUTHOR]

Paschke's version of Stinespring's theorem associates a Hilbert C ∗ -module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a C ∗ -algebra 𝒜 one may associate an inclusion system E = (E t) of Hilbert 𝒜 - 𝒜 -modules with a generating unit ξ = (ξ t). Suppose ℬ is a von Neumann algebra, consider M 2 (ℬ) , the von Neumann algebra of 2 × 2 matrices with entries from ℬ. Suppose (Φ t) t ≥ 0 with Φ t = ϕ t 1 ψ t ψ t ∗ ϕ t 2 , is a QDS on M 2 (ℬ) which acts block-wise and let (E t i) t ≥ 0 be the inclusion system associated to the diagonal QDS (ϕ t i) t ≥ 0 with the generating unit (ξ t i) t ≥ 0 , i = 1 , 2. It is shown that there is a contractive (bilinear) morphism T = (T t) t ≥ 0 from (E t 2) t ≥ 0 to (E t 1) t ≥ 0 such that ψ t (a) = 〈 ξ t 1 , T t a ξ t 2 〉 for all a ∈ ℬ. We also prove that any contractive morphism between inclusion systems of von Neumann ℬ - ℬ -modules can be lifted as a morphism between the product systems generated by them. We observe that the E 0 -dilation of a block quantum Markov semigroup (QMS) on a unital C ∗ -algebra is again a semigroup of block maps. [ABSTRACT FROM AUTHOR]