PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 51, Number 1, August 1975
COMPACTOPERATORSIN THE ALGEBRAGENERATEDBY ESSENTIALLYUNITARY CQ OPERATORSl ERIC A. NORDGREN ABSTRACT. It will be shown that the compact operators in the weakly closed algebra generated by an essentially unitary C contraction are weakly dense in the algebra. The result implies the extension of a double dual theorem of Kriete, Moore and Page and yields a partial answer to a question on reductive algebras raised
by Rosenthal.
A contraction tary in case Muhly's
T on a separable
both 1 — iT
characterization
erator,
was introduced
generated tains
nonzero
and 1 ~ Ti
ate compact. operators
in [5] for showing
compact
of a sequence
unitary
operators
A technique,
uni-
based
on
with a C0(. op-
closed
algebra
'S.j.
T and the identity
technique
operator
essentially
commuting
contraction
That
every
is called
that the weakly
C
operators.
and hence
of compact
space
[6] of compact
by an essentially
that the identity,
Hilbert
will be used
in 21^,, is in fact
here
con-
to show
the weak
in =
\(f> iT)\
by regularity that
lim
sequence
converges
according
determines
n terms
to multiplicity,
\ converges
factor
of T of Lebesgue
measure
of v, there
a sequence
viK
77—.oo
n
exists
) = i/(F 0J. y
72
is continuous
D.
weakly
the singular
of ¡a, \ and whose
n
of such
III. 2.1. c ]).
repeated that
diviin [5]
whose
zero,
are the first
\y
inner
of mj.
measure
D K ), and let
that
As
of the open unit disc
point
measure
2IT be
It is shown
divisor
(even
' n
analytic)
For each
n
be the inner
singular
factor
on T\fC
to 7zzT on D, and hence
ifi
analytic
on a closed
343—345] such
we will recall
function
that
Fatou
a portion
on the disc subset
first
of T having
constructs
is constructed
continuous
for each
limit
t.
func-
of E, the harradial
Thus if xfi= exp — ih + ik), then
=-(f>it)
Taking
then obtains
of
