By Mahowald M., Priddy S. (eds.)
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If there exists D ∈ B(H) such that DD∗ = 1 − Td Td∗ and D commutes with T then T has a unitary dilation. Proof. Firstly consider the case when Td is a coisometry. There exists a positive kernel K for T. Fix α in Zd−1 . Deﬁne Kj (α) ∈ B(H), j = 0, 1, . . , by Kj (α) = Tdj K(α)Tdj∗ . Note that Kj (α) ≤ K(α) for all j, and hence b Kj (α) exists. Call this K(α). In this way we deﬁne K : Zd−1 → B(H) that we claim is also a positive kernel for T. To see this, note that then (a) if α is in Zd−1 + K(α) = b Tdj K(α)Tdj∗ = b T α Tdj Tdj∗ = b(T α ) = T α ; and (b) K(α − β)x(β), x(α) α,β∈Zd−1 + K(α − β)Tdj∗ x(β), Tdj∗ x(α) =b ≥0 (since the term in the square brackets is positive for each j).
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A) There exists a Hilbert space M and a unitary dilation U of T on M ⊕ H ⊕ D such that each U α has the form ⎞ ⎛ ∗ ∗ ∗ ⎝0 T α Dα ⎠ , α ∈ Zd+ . 0 0 ∗ (b) There exists a commuting coisometric extension W of T on H ⊕ D such that each W α has the form Tα 0 Dα , ∗ α ∈ Zd+ . 10) for all α, β ∈ Zd+ . 11) is a positive kernel for T. Proof. (a)⇒(b): Deﬁne W by setting Wj = PH⊕D Uj |H⊕D , j = 1, . . , n. It is straightforward to verify that W has the required properties. (b)⇒(c): Clearly D0 = PH 1|D = 0.
Algebraic Topology by Mahowald M., Priddy S. (eds.)