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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Robert T. Powers and Geoffrey Price
Journal: Trans. Amer. Math. Soc. 321 (1990), 347-361
MSC: Primary 47D25; Secondary 46L99, 47D05
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0974524-0
PII: S 0002-9947(1990)0974524-0
Article copyright: © Copyright 1990 American Mathematical Society