I nternat. J. Math. & Math. Si.
Vol.
(1978)93-96
A NOTE ON RIESZ ELEMENTS
IN C*-ALGEBRAS
D AVID LEGG
Department of Mathematics
Indiana Un iversity-Purdue University
Fort Wayne, Indiana
(Received December 5, 1977)
ABSTRACT.
written R
It is known that every Riesz operator R on a Hilbert space can be
Q + C, where C is compact and both Q and CQ
QC are quasinilpotent.
This result is extended to a general C*-algebra setting.
INTRODUCTION.
In [3], Smyth develops a Riesz theory for elements in a Banach algebra with
respect to an ideal of algebraic elements.
In [I], Chui, Smith and Ward show
that every Riesz operator on a Hilbert space is decomposible into R
where C is compact and both Q and CQ
QC are quasinilpotent.
Q + C,
In this paper we
use Smyth’s work to show that the analogous result holds in an arbi#rary
C*-a Igebra.
2.
DEFINITIONS AND NOTATION.
Let A be a C*-algebra, and let F be a two-sided ideal of algebraic elements
D. LEGG
94
An element T
of A.
A is a Riesz element if its coset T + F in A/F has
(T) is a finite pole of T f ft is isolated in
A point Z
spectral radius O.
(T) and the corresponding spectral projection lies in F.
is not a finite pole of T}.
Smyth also showed that if T is a Riesz
Q + U, where Q i quasinilpotent and U
This is a generalization of
F.
West’s result [4, Thm. 7.5].
[3, Thm. 6.9].
We now extend the
result of Chui, Smith and Ward [I, Thm. I] by showing that UQ
QU is quasinil-
Q + U is the Smyth decomposition.
potent, where T
3.
(T)’Z
Smyth has shown that T is a Riesz element if and
only if E(T) {0}, [3, Thm. 5.3].
element, then T
{X
Let E(T)
OUTLINE OF SMYTH’S CONSTRUCTION.
Let T be a Riesz element, and label the elements of (T)\E(T) by
n
I, 2,
n
in such a way that
In
>
.IXn+
x n +0
finite pole, so each spectral projection P is in F
n
then fnd a self-adjont projection
Qn
Let V n
Qn-1’
Let S n
Qnsatisfying SnQ n
and define U
as n.
Qn
and
P
Each X
+
+ P
QnSn
n
S n.
U is clearly in F and Q
kVk
n is a
T -U is
shown to be quasinilpotent.
4.
THEOREM
PROOF.
UQ
QU is quasinilpotent.
For any S s A, let S denote the left regular representatfon of S.
Then by Lemma 6.6 in Smyth [3], we have that
Since
S
Qv
A
Qn
QnQn
That is,
QQnX QnSX
subspace of Q.
form
we have
QQn
Qn
Qn S
e
Qn A"
Hence
Now let v
belongs to range
Qn"
Qn A is an
Q(Qn e Qn A,
range
Qn’
nvariant subspace of
say
say v
Qn S
Q(Qn
Qn x
Hence we see that range
.
for some
Then
Qn
is an
nvariant
It follows that Q has an operator matrix representation of the
RIESZ ELEMENTS IN
A22 A23
0
where
Aij
A 13
A 12
A 11
C*-ALGEBRAS
With
ViQV j.
0
,212
0
0
respect to
this blocking, we have
0
X313
0
(1-:kn)A1n
0
He n ce
0
o
O,n_1-,n )A n-l,n
uQ
Qu
95
96
D. LEGG
Now let P be the orthogonal projection onto
range Qn’
and let
n
An
Qn)(UQ
QU)(P
as n.
Hence UQ
QU
But UQ
QU
(P
Qn ).
It Is easy to see thee
IA n l IXnlll
A n converges In the un!form norm to UQ
QU as n.
A n has the form
where N is nilpotent.
It follows that UQ
QU
A n has no non-zero e!genvalues.
,
Thm. 3.], p. 14 of [2] can now be easily modified to show that UQ
non-zero eigenvalues.
i.e., UQ
diag.
Since UQ
QU be longs to
this means a(UQ
QU has no
QU)
{0},
QU is quasinilpotent.
REFERENCES
I.
Chui, C. K., Smith, P. W., and Ward, J. D., A note on R!esz operators,
Amer. Math. Soc., 60, (1976), 92-94.
2.
Gohberg, [. Co, and Krein, M. G., Introduction to the theory of linear nonselfadJoint operators. "Nauka," Moscow, 1965; English transl., Transl.
Math Monographs, vol. ]8, Amer. Math. Soc., Providence, R. I. 1969.
3.
Syth, M. R. F., Riesz theory in Banach algebras, Math Z., ]45, (1975),
45-155.
4.
West, T. T., The decomposition of Riesz operators, Proc. London Math. Soc.,
]IT, Ser. 16, (966), 737-752.
Proc_____=.
AMS (MOS) Subject Classification numbers 47 B 05, 47 C 10
KEY WORDS AND PHRASES.
C*
algebra, quasiilpote
op,
Ritz
n.