Continuous spatial semigroups of $*$-endomorphisms of $\mathfrak {B}(\mathfrak {H})$

Abstract:

To each continuous semigroup of $*$-endomorphisms $\alpha$ of $\mathfrak {B}\left ( \mathfrak {H} \right )$ with an intertwining semigroup of isometries there is associated a $*$-representation $\pi$ of the domain $\mathfrak {O}(\delta )$ of the generator of $\alpha$. It is shown that the Arveson index ${d_ * }(\alpha )$ is the number of times the representation $\pi$ contains the identity representation of $\mathfrak {O}(\delta )$. This result is obtained from an analysis of the relation between two semigroups of isometries, $U$ and $S$, satisfying the condition $S{(t)^*}U(t) = {e^{ - \lambda t}}I$ for $t \geq 0$ and $\lambda > 0$.