COMPACT OPERATORS IN THE ALGEBRA GENERATED BY

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ABSTRACT. It will be shown that the compact operators in the weakly closed algebra generated by an essentially unitary. C con- traction are weakly dense.
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