Lemma: Averaged quadratic form

For $S\in \mathcal{B}(H)$, define $P(S)$ by

Then $P$ is a positive unital contraction, $P(S)$ commutes with every spectral projection of $D$, and

In particular,

No normality of $P$ is asserted.

Proof. The bounded sesquilinear form defines $P(S)$ with norm at most $\Vert S\Vert$. Positivity and unitality follow from the mean. Translation invariance gives $U_u^{\ast }P(S)U_u=P(S)$, hence spectral commutation.

For $L(a)$, the conjugated operators belong to the weakly closed algebra $L(M)$. Their averaged weak limit belongs to $L(M)$ and commutes with the modular group, so is scalar. Evaluation at $\xi$ gives $\phi (a)I_H$. Equation (eq:averagedform) is the definition for $T^{\ast }T$.