Estimation optimale du gradient du semi-groupe de la chaleur sur le ...

466 H. Schultz / Journal of Functional Analysis 236 (2006) 457–489
Theorem 2.9. [2] Let E and F be (not necessarily closed) subspaces of H which are affiliated
with M. 2 Then E ∩ F is affiliated with A, and
E ∩ F = E ∩ F. (2.25)
Lemma 2.10. Consider idempotents e, f ∈ ˜M. Let P = Prange(e), Q = Prange(1−e), R = Prange(f )
and S = Prange(1−f). Then ef = feif and only if
Proof. Clearly,
(P ∧ R) ∨ (P ∧ S) ∨ (Q ∧ R) ∨ (Q ∧ S) = 1. (2.26)
1 = ef + e(1 − f)+ (1 − e)f + (1 − e)(1 − f). (2.27)
Suppose that ef = fe, and let g1 = ef , g2 = e(1 − f), g3 = (1 − e)f and g4 = (1 − e)(1 − f).
Then g1,...,g4 are idempotents with support projections P1, P2, P3 and P4, respectively, such
that
4
(H) = H. (2.28)
Pi
i=1
Moreover, P1 P ∧ R, P2 P ∧ S, P3 Q ∧ R and P4 Q ∧ S. This shows that (2.26) holds.
On the other hand, assume that (2.26) holds. According to (2.26) and Theorem 2.9,
H0 := D(ef ) ∩ D(f e) ∩ (P ∧ R)(H) + (P ∧ S)(H) + (Q ∧ R)(H) + (Q ∧ S)(H)
is dense in H so it suffices to prove that ef and fe agree on H0. To see this, let ξ ∈ D(ef ) ∩
D(f e) ∩ (P ∧ R)(H). Then
ef ξ = eξ = ξ = fξ = feξ, (2.29)
and similarly, when ξ ∈ D(ef ) ∩ D(f e) ∩ (P ∧ S)(H), ξ ∈ D(ef ) ∩ D(f e) ∩ (Q ∧ R)(H) or
ξ ∈ D(ef ) ∩ D(f e) ∩ (Q ∧ S)(H). Thus, ef agrees with feon H0. ✷
3. An idempotent valued measure associated with T ∈ M
As in the previous section, consider a finite von Neumann algebra M with a faithful, normal,
tracial state τ . Inspired by the notion of a spectral measure we make the following definition.
Definition 3.1. Let (X, F) denote a measurable space. An idempotent valued measure on (X, F)
(with values in ˜M) isamape from F into I( ˜M) such that:
(i) e(X) = 1,
(ii) e(F1)e(F2) = e(F2)e(F1) = 0 when F1,F2 ∈ F with F1 ∩ F2 =∅,
2 A subspace E of H is said to be affiliated with M if for all T ∈ M ′ , T(E)⊆ E. Note that if E is affiliated with M,
then the projection onto E belongs to M,andifE is closed, then this is a necessary and sufficient condition for E to be
affiliated with M.