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

E. Kissin / Journal of Functional Analysis 236 (2006) 609–629 629
Hence UAU ∗ ∈ CS, soUCSU ∗ = CS.AsJS is an ideal of AS,
UD(δ)U ∗ = U(CS + JS)U ∗ ⊆ D(δ).
Thus U ∈ Z(δ), soZS ⊆ Z(δ). Since always Z(δ) ⊆ ZS, wehaveZ(δ) = ZS. ✷
Let S be a selfadjoint operator on H = ∞ i=−∞ Hi and let S|Hi = λi1Hi with all distinct λi.
The group ΓS is described in Corollary 5.2 and CS consists of bounded operators commuting
with all Pλi . By Theorem 6.3 and Proposition 6.4, δ = δS|(CS + JS) is a closed *-derivation of
the C*-algebra CS + C(H) and Z(δ) = ZS.
Acknowledgment
We are very grateful to Victor Shulman for his valuable suggestions about this paper.
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