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By Aitken R.J.

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Extra resources for A Statistical Study of the Visual Double Stars in the Northern Sky (1915)(en)(5s)

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1I ⊗ R)n−1 (˜ u1 ) = w∗ (1I ⊗ R)(w) . . (1I ⊗ R)n−1 (w∗ )(1I ⊗ R)n (w) = w∗ (1I ⊗ R)n (w). Proposition: J˜ and (1I ⊗ R)| ˜ q are conjugate if and only if u1 can be q B(H) chosen in such a way that (w ˜n )∞ n=1 is a coboundary (with projection q). Proof: If u˜1 = w ˜1 = w∗ (1I ⊗ R)(w), then ˜ x) = u J(˜ ˜1 (1I ⊗ R)(˜ x) u˜∗1 = w∗ (1I ⊗ R)(w x˜ w∗ )w, proving conjugacy. e. u (1I ⊗ R)(˜ x) for all x ˜ ∈ B(H), ˜∗1 w∗ (1I ⊗ R)(w) = y ∈ B(K1 ). The modified † u1 = u1 y generates the same endomorphism J˜ (because yy ∗ = 1I) and satisfies u†1 = w∗ (1I ⊗ R)(w).

RN . N →∞ To get a product representation in the general case we can embed Dn and Dn∗ into a Hilbert space Kn for all n, in such a way that Rn+1 has a uni˜ n+1 on Kn ⊕ Kn+1 . The construction of the rotation matrices tary extension R ∞ shows that it is possible to take a copy of H0 for each Kn . On n=0 Kn we can proceed as in the one dimensional case and we get ˜ := Corollary: In the setting of the proposition define K0 = H0 and H ∞ n m=0 Km , where each Km is a copy of H0 , further K[0,n] := m=0 Km .

Using Ad(u1 )(A ⊗ C) = A ⊗ C we find that A˜ is globally invariant R)n−1 (˜ stop for Ad(wn ) and Ad(wn∗ ). 6(5) we know that wn∗ → w ˜ (with q = q[1,∞) , the projection and the range of w is equal to K[1,∞) = q H stop ∗ ˜ onto K[1,∞) ). 6), from H n wop wn → w∗ . The algebra C[1,∞) as represented on K[1,∞) may be identified with ˜ Therefore we get for all a q Aq. e. e. Ad(w∗ )(C[1,∞) ) ⊂ A. Here the assumption has been used that A˜ and C[1,∞) are closed in the strong and weak operator topologies.

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