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Summary: THE E0-SEMIGROUP SENTINEL
VOLUME 1, NUMBER 1
Bill Arveson, Editorial Staff
13 April 1999
The purpose of these remarks is to report some news concerning type III E0-
semigroups. I've been working through Bob Powers' remarkable paper [1], hoping
to understand his construction in terms as simple as possible and to make connec-
tions with other parts of operator theory (like Wiener-Hopf operators and Hankel
operators) where that seemed helpful. The following discussion summarizes what I
have learned, and appears to clarify things to some extent.
For notation, ^R will denote the dual group of the additive group R of real num-
bers, and I choose normalizations of Lebesgue measure on R and ^R which cause
irritating factors of
2 and its reciprocal, that put in unwanted appearances when
taking Fourier transforms, to disappear. This causes the unit interval to have pe-
culiar length; but that problem is secondary since we do no numerical calculations.
Following the physicists, I will remind you of where we are by using the letters x, y
for elements of R and p, q ^R for elements of ^R. C(^R) will denote all continuous
functions f : ^R C which have a limit at the point at ,
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