4-Manifold topology I: Subexponential groups by Freeadman M.H., Teichner P.

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By Freeadman M.H., Teichner P.

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Thus we have QT = W Q. 3) Let us now consider the space H of vectors h = {hn }∞ −∞ with hn ∈ DT and h 2 ∞ = ∑ hn −∞ 2 < ∞. We embed H in K = H ⊕ Q by identifying the element h of H with the element {. . , 0, 0, DT h , DT T h, DT T 2 h, . 2). ) 10. A NOTHER METHOD TO CONSTRUCT ISOMETRIC DILATIONS 39 Let V be the bilateral shift on H defined by V {hn} = {h′n }, where h′n = hn+1 (n = 0, ±1, . 3) it follows for h ∈ H and m = 1, 2, . . 4) n=−∞ ⊕ 0, where (m) hn DT T m+n h 0 = if −m ≤ n ≤ −1, in the other cases.

T x)n ) = D(x) D(x1 , . . , ∂xj ∂xj ( j = 1, . . , n) of the system of linear differential equations n ∂ Xi (τt x) dui = ∑ uk dt k=1 ∂ xk (i = 1, . . 12), and hence one derives, using Liouville’s theorem and the fact that δ0 (x) = 1, the formula δt (x) = exp − t 0 ρ (τs x)ds . 13) Thus for any Borel-measurable function ϕ (x) integrable on Rn we have Rn ϕ (τ−t x) dx = Rn ϕ (x)δt (x) dx (dx = dx1 . . dxn ). 14) Set, for f (x) ∈ L2 (Rn ) and for t ≥ 0, (T (t) f )(x) = f (τ−t x). 14) imply that T (t) is a contraction of L2 (Rn ) for t ≥ 0.

For continuous f (x) with compact support it is obvious that T (t) f → f strongly, as t → +0. These functions being dense in L2 (Rn ) we conclude easily that the semigroup {T (t)} is continuous. Let us introduce the measures d ν (x, s) = ρ (x)dxds (in Rn+1 ), where ∞ δ∞ (x) = exp − 0 d µ (x) = δ∞ (x)dx (in Rn ), ρ (τs x)ds = lim δt (x) (≥ 0). 1. 12) with divergence −ρ (x) ≤ 0, has an isometric dilation that is unitarily equivalent to the semigroup {U(t)}t≥0 defined on the space K = L2 (Rn+1 ; ν ) ⊕ L2 (Rn ; µ ) by U(t)[f(x, s) ⊕ f (x)] = f(x, s + t) ⊕ f (τ−t x).

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