 
Summary: THE E0SEMIGROUP 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 WienerHopf 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 ,
