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Measurable and Continuous Units of an $E_{0}$-semigroup

Published online by Cambridge University Press:  25 March 2020

S. P. Murugan
Affiliation:
Indian Institute of Science, Education and Research, Mohali, 140306, Punjab, India Email: [email protected]
S. Sundar
Affiliation:
Institute of Mathematical Sciences (HBNI), Taramani, 600113, Tamilnadu, India Email: [email protected]

Abstract

Let $P$ be a closed convex cone in $\mathbb{R}^{d}$ which is spanning, i.e., $P-P=\mathbb{R}^{d}$ and pointed, i.e., $P\,\cap -P=\{0\}$. Let $\unicode[STIX]{x1D6FC}:=\{{\unicode[STIX]{x1D6FC}_{x}\}}_{x\in P}$ be an $E_{0}$-semigroup over $P$ and let $E$ be the product system associated to $\unicode[STIX]{x1D6FC}$. We show that there exists a bijective correspondence between the units of $\unicode[STIX]{x1D6FC}$ and the units of $E$.

Type
Article
Copyright
© Canadian Mathematical Society 2019

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References

Anbu, Arjunan, Srinivasan, R., and Sundar, S., E-semigroups over closed convex cones. arxiv:math.OA:1807.11375.Google Scholar
Arveson, William, Noncommutative dynamics and E-semigroups. Springer Monogr. Math., Springer-Verlag, New York, 2003.10.1007/978-0-387-21524-2CrossRefGoogle Scholar
Bargmann, V., On unitary ray representations of continuous groups. Ann. of Math. (2) 59(1954), 146.10.2307/1969831CrossRefGoogle Scholar
Faraut, Jacques and Koranyi, Adam, Analysis on symmetric cones. Oxford Math. Monogr., Oxford University Press, New York, 1994.Google Scholar
Laca, Marcelo and Raeburn, Iain, Extending multipliers from semigroups. Proc. Amer. Math. Soc. 123(1995), 355362.10.1090/S0002-9939-1995-1227519-6CrossRefGoogle Scholar
Murugan, S. P. and Sundar, S., E 0P-semigroups and product systems. arxiv:math.OA:1706.03928.Google Scholar
Palmer, Theodore W., Banach algebras and the general theory of ∗-algebras, Vol. 2. Encyclopedia Math. Appl. Vol. 79, Cambridge University Press, Cambridge, 2001.10.1017/CBO9780511574757CrossRefGoogle Scholar
Varadarajan, V. S., Geometry of quantum theory. second ed., Springer-Verlag, New York, 1985.Google Scholar