Algebre: Chapitre 10.Algebre homologique by N. Bourbaki

By N. Bourbaki

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce dixième chapitre du Livre d Algèbre, deuxième Livre du traité, pose les bases du calcul homologique.

Ce quantity est a été publié en 1980.

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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.

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