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

HTML articles powered by AMS MathViewer

- by Robert T. Powers and Geoffrey Price PDF
- Trans. Amer. Math. Soc.
**321**(1990), 347-361 Request permission

## 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 $\lambda > 0$.## References

- William Arveson,
*An addition formula for the index of semigroups of endomorphisms of $B(H)$*, Pacific J. Math.**137**(1989), no. 1, 19–36. MR**983326**, DOI 10.2140/pjm.1989.137.19 - William Arveson,
*Continuous analogues of Fock space*, Mem. Amer. Math. Soc.**80**(1989), no. 409, iv+66. MR**987590**, DOI 10.1090/memo/0409 - Ola Bratteli, Richard H. Herman, and Derek W. Robinson,
*Perturbations of flows on Banach spaces and operator algebras*, Comm. Math. Phys.**59**(1978), no. 2, 167–196. MR**491612**, DOI 10.1007/BF01614248 - R. G. Douglas,
*On the $C^{\ast }$-algebra of a one-parameter semigroup of isometries*, Acta Math.**128**(1972), no. 3-4, 143–151. MR**394296**, DOI 10.1007/BF02392163
N. Dunford and J. T. Schwartz, - Robert T. Powers,
*An index theory for semigroups of $^*$-endomorphisms of ${\scr B}({\scr H})$ and type $\textrm {II}_1$ factors*, Canad. J. Math.**40**(1988), no. 1, 86–114. MR**928215**, DOI 10.4153/CJM-1988-004-3 - Robert T. Powers,
*A nonspatial continuous semigroup of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$*, Publ. Res. Inst. Math. Sci.**23**(1987), no. 6, 1053–1069. MR**935715**, DOI 10.2977/prims/1195175872 - Robert T. Powers and Derek W. Robinson,
*An index for continuous semigroups of $*$-endomorphisms of ${\mathfrak {B}}({\mathfrak {H}})$*, J. Funct. Anal.**84**(1989), no. 1, 85–96. MR**999489**, DOI 10.1016/0022-1236(89)90111-0 - Michael Reed and Barry Simon,
*Methods of modern mathematical physics. I. Functional analysis*, Academic Press, New York-London, 1972. MR**0493419** - Frigyes Riesz and Béla Sz.-Nagy,
*Functional analysis*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR**0071727** - Derek W. Robinson,
*The approximation of flows*, J. Functional Analysis**24**(1977), no. 3, 280–290. MR**0440420**, DOI 10.1016/0022-1236(77)90059-3 - Béla Sz.-Nagy,
*Isometric flows in Hilbert space*, Proc. Cambridge Philos. Soc.**60**(1964), 45–49. MR**159231**, DOI 10.1017/S0305004100037427 - Kôsaku Yosida,
*Functional analysis*, 4th ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag, New York-Heidelberg, 1974. MR**0350358**, DOI 10.1007/978-3-642-96208-0

*Linear operators*, Interscience, New York, 1958; reprinted 1971.

## Additional Information

- © Copyright 1990 American Mathematical Society
- 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