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We give formal definitions as follows. 1. 1. Direct sum: Let E and F be two (A, B)-correspondences. Then the directsum correspondence E ⊕ F is defined to be the direct sum vector space E ⊕ F together with the diagonal left-A action and right-B action and the directsum B-valued inner product: a · (e ⊕ f ) = (a · e) ⊕ (a · f ), e ⊕ f, e ⊕ f E⊕F = e, e E (e ⊕ f ) · b = (e · b) ⊕ (f · b), + f, f F. 2. Tensor product: Suppose that we are given three C ∗ -algebras A, B and C together with an (A, B)-correspondence E and a (B, C)-correspondence F .

5), we may write ϕ(n) (a) = ϕ(a) ⊗ IE ⊗n−1 . We formally set E ⊗0 = A. 1) and is also an A-correspondence. A. Ball, A. Biswas, Q. Fang and S. ter Horst or, more succinctly, ϕ∞ (a) = diag(a, ϕ(1) (a), ϕ(2) (a), . ). In addition to the von Neumann algebra A and the A-correspondence E, suppose that we are also given an auxiliary Hilbert space E and a nondegenerate ∗-homomorphism σ : A → L(E); as this will be the setting for much of the analysis to follow, we refer to such a pair (E, σ) as a correspondence-representation pair.

The notation is mostly standard but we mention here a few conventions for reference. For Ω any index set, 2 (Ω) denotes the space of complex-valued functions on Ω which are absolutely square summable: 2 (Ω) = {ξ : Ω → C : |ξ(ω)|2 < ∞}. ω∈Ω Most often the choice Ω = Z (the integers) or Ω = Z+ (the nonnegative integers) appears. For H a Hilbert space, we use 2H (Ω) as shorthand for 2 (Ω) ⊗ H (the space of H-valued function on Ω square-summable in norm). More general versions where H may be a correspondence also come up from time to time.

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A Carleman Function and the Cauchy Problem for the Laplace Equation by Yarmukhamedov Sh.


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