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

640 S. Albeverio, A. Kosyak / Journal of Functional Analysis 236 (2006) 634–681
3. Idea of the proof of irreducibility
Proof of Theorem 5. The proof of Theorem 5 is organized as follows:
(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i).
The parts (i) ⇒ (ii) ⇒ (iii) are evident. The part (iii) ⇔ (iv) follows from Lemma 8, which is
based on the Kakutani criterion [15].
The idea of the proof of irreducibility, i.e. the part (iv) ⇒ (i). Let us denote by A m the von
Neumann algebra generated by the representation T R,μm B
A m = T R,μm B
t
| t ∈ G ′′ .
We show that (iv) ⇒[(Am ) ′ ⊂ L∞ (Xm ,μm B )]⇒(i). Let the inclusion (Am ) ′ ⊂ L∞ (Xm ,μm B )
holds. Using the ergodicity of the measure μm B (Lemma 6) this proves the irreducibility. Indeed
in this case an operator A ∈ (Am ) ′ should be the operator of multiplication (since (Am ) ′ ⊂
L∞ (Xm ,μm B )) by some essentially bounded function a ∈ L∞ (Xm ,μm B ). The commutation relation
[A,T R,μm B
t ]=0 ∀t ∈ BN 0 implies a(R−1 t (x)) = a(x) (mod μm B ) ∀t ∈ BN 0 , so by ergodicity of
the measure μ m B with respect to the right action of the group BN 0 on the space Xm we conclude
that A = a = const (mod μm B ). This then proves the irreducibility in Theorem 5, i.e. the part
[(Am ) ′ ⊂ L∞ (Xm ,μm B )]⇒(i).
The proof of the remaining part, i.e. the implication (iv) ⇒[(Am ) ′ ⊂ L∞ (Xm ,μm B )] is based
on the fact that the operators of multiplication by independent variables xpq, 1 p m, p