Completely Dissipative Maps and Stinespring ’ s Dilation - Type Theorem on σ - C ∗ - Algebras

and Applied Analysis 3 From the corresponding facts in the Aronszajn-Kolmogorov theorem of 17 , we have the following result. Lemma 2.3 see 17 . Let A be a σ-C∗-algebra,H a Hilbert space, and a map T : A ×A → B H be given such that, for all a1, a2, . . . , an ∈ A, [ T ( ai , aj )]n i,j 1 ∈ M n B H . 2.3 Then there exists a Hilbert space K and a map V : A → B H,K such that T a, b V a∗ ∗V b , ∀a, b ∈ A. 2.4 Definition 2.4. Let A be a σ-C∗-algebra,H a Hilbert space, and a map φ : A ×A ×A → B H . We say that φ is completely positive definite if for any ai, bi ∈ A, i 1, 2, . . . , n ∈ N, we have [ φ ( b∗ i , a ∗ i aj , bj )]n i,j 1 ∈ M n B H . 2.5 If there exists a function ρ : A → R such that the map φ : A × A × A → B H satisfies the following additional condition: n ∑ i 1 n ∑ j 1 〈 φ ( b∗ i , a ∗ i a ∗aaj , bj ) xj , xi 〉 ≤ ρ a∗a n ∑ i 1 n ∑ j 1 〈 φ ( b∗ i , a ∗ i aj , bj ) xj , xi 〉 , 2.6 for every a, a1, a2, . . . , an, b1, b2, . . . , bn ∈ A, x1, x2, . . . , xn ∈ H, and n ∈ N, then we say, φ is relatively bounded. 3. The Stinespring’s Dilation-Type Theorem To prove the main theorem, the following things have to be elucidated. If X and Y are vector spaces, we denote by X ⊗ Y their algebraic tensor product. This is linearly spanned by elements x ⊗ y x ∈ X, y ∈ Y . If σ : X × Y → Z is a bilinear map, where X,Y , and Z are vector spaces, then there is a unique linear map σ ′ : X ⊗ Y → Z such that σ ′ x ⊗ y σ x, y for all x ∈ X and y ∈ Y . If τ, μ are linear functionals on the vector spaces X,Y , respectively, then there is a unique linear functional τ ⊗ μ on X ⊗ Y such that ( τ ⊗ μx ⊗ y τ x μy, ∀x ∈ X, y ∈ Y, 3.1 since the function X × Y −→ C, x, y −→ τ x μy 3.2