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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Robert T. Powers and Geoffrey Price
Journal: Trans. Amer. Math. Soc. 321 (1990), 347-361
MSC: Primary 47D25; Secondary 46L99, 47D05
DOI: https://doi.org/10.1090/S0002-9947-1990-0974524-0
MathSciNet review: 974524
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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 $\lambda > 0$.


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Article copyright: © Copyright 1990 American Mathematical Society