# Algebre: Chapitre 10.Algebre homologique by N. Bourbaki By N. Bourbaki

Ce dixiÃ¨me chapitre du Livre d AlgÃ¨bre, deuxiÃ¨me Livre du traitÃ©, pose les bases du calcul homologique.

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Additional info for Algebre: Chapitre 10.Algebre homologique

Example text

If (Pi ) is an increasing net of projections, then Pi → i Pi strongly; if (Pi ) is decreasing, then Pi → i Pi strongly. Note that L(H)+ is not a lattice unless H is one-dimensional. 1 0 1 1 For example, P = and Q = 12 have no least upper bound 0 0 1 1 1 0 3 1 in (M2 )+ . e. ] Thus the supremum of P and Q in Proj(H) is not necessarily a least upper bound of P and Q in L(H)+ . However, if (Pi ) is an increasing net of projections in L(H), then P = Pi is the least upper bound for {Pi } in L(H)+ , since Pi → P strongly, and for each T ∈ L(H)+ , {S ∈ L(H) : 0 ≤ S ≤ T } is strongly closed.

Thus there is polar decomposition for closed operators. 6, polar decomposition also works for closed conjugatelinear operators. 6). 2, then |T0 | has dense range since T0 is one-to-one, but |T1 | does not have dense range. Finally, there is a more nontrivial application of functional calculus which is very important. Let H be a self-adjoint operator on H, and for t ∈ R set Ut = eitH . Then Ut is unitary, Us+t = Us Ut , and t → Ut is strongly continuous ((Ut ) is a strongly continuous one-parameter group of unitaries).

The projections P and Q are called the initial and ﬁnal projections, or source and range projections, of U . 4 Proposition. Let S, T ∈ L(H) with S ∗ S ≤ T ∗ T . Then there is a unique W ∈ L(H) with W ∗ W ≤ QT (hence W ≤ 1), and S = W T . If R ∈ L(H) commutes with S, T , and T ∗ , then RW = W R. Proof: W is deﬁned on R(T ) by W (T ξ) = Sξ (W is well deﬁned since Sξ ≤ T ξ for all ξ). W extends to an operator on QT H by continuity; set ∗ ∗ ∗ W = 0 on Q⊥ T H. Then W W ≤ QT and S = W T . 6 The Spectral Theorem 23 and thus RW η = W Rη for η ∈ QT H.