anoteon in c*-algebrasdownloads.hindawi.com/journals/ijmms/1978/169842.pdf · internat.j. math....

I nternat. J. Math. & Math. Si.Vol. (1978)93-96

A NOTE ON RIESZ ELEMENTSIN C*-ALGEBRAS

DAVID LEGGDepartment of Mathematics

Indiana Un iversity-Purdue UniversityFort Wayne, Indiana

(Received December 5, 1977)

ABSTRACT. It is known that every Riesz operator R on a Hilbert space can be

written R 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 Q + C,

where C is compact and both Q and CQ QC are quasinilpotent. 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

94 D. LEGG

of A. An element T A is a Riesz element if its coset T + F in A/F has

spectral radius O. A point Z (T) is a finite pole of T f ft is isolated in

(T) and the corresponding spectral projection lies in F. Let E(T) {X (T)’Z

is not a finite pole of T}. Smyth has shown that T is a Riesz element if and

only if E(T) {0}, [3, Thm. 5.3]. Smyth also showed that if T is a Riesz

element, then T Q + U, where Q i quasinilpotent and U F. [3, Thm. 6.9].

This is a generalization of West’s result [4, Thm. 7.5]. We now extend the

result of Chui, Smith and Ward [I, Thm. I] by showing that UQ QU is quasinil-

potent, where T Q + U is the Smyth decomposition.

3. OUTLINE OF SMYTH’S CONSTRUCTION.

Let T be a Riesz element, and label the elements of (T)\E(T) by

n n I, 2, in such a way that I > .IXn+ x +0 as n. Each X is an n n

finite pole, so each spectral projection P is in F Let S P + + Pn n n

then fnd a self-adjont projection Qnsatisfying SnQn Qn and QnSn Sn.

Let Vn Qn Qn-1’ and define U kVk U is clearly in F and Q T -U is

shown to be quasinilpotent.

4. THEOREM UQ QU is quasinilpotent.

PROOF. 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 QnA is an nvariant subspace of .Since Qn QnQn we have Qn e QnA" Hence Q(Qn e QnA, say Q(Qn Qn S for some

QQn Qn say v Qn x ThenS A That is, S Now let v range Qn’Qv QQnX QnSX belongs to range Qn" Hence we see that range Qn is an nvariant

subspace of Q. It follows that Q has an operator matrix representation of the

form

RIESZ ELEMENTS IN C*-ALGEBRAS 95

A A A11 12 13

0 A22 A23

where Aij ViQVj. With respect to this blocking, we have

0 ,212 0

0 0 X313

0

Hence

uQ Qu

0

0 o(1-:kn)A1n

O,n_1-,n )An-l,n

96 D. LEGG

Now let P be the orthogonal projection onto range Qn’ and letn

A (P Qn)(UQ QU)(P Qn ). It Is easy to see thee IA l IXnlll diag.n n

as n. Hence UQ QU An converges In the un!form norm to UQ QU as n.

But UQ QU A has the formn

where N is nilpotent. It follows that UQ QU An has no non-zero e!genvalues.

Thm. 3.], p. 14 of [2] can now be easily modified to show that UQ QU has no

non-zero eigenvalues. Since UQ QU be longs to , this means a(UQ QU) {0},

i.e., UQ QU is quasinilpotent.

REFERENCES

I. Chui, C. K., Smith, P. W., and Ward, J. D., A note on R!esz operators, Proc_____=.Amer. Math. Soc., 60, (1976), 92-94.

2. Gohberg, [. Co, and Krein, M. G., Introduction to the theory of linear non-selfadJoint 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.

AMS (MOS) Subject Classification numbers 47 B 05, 47 C 10

KEY WORDS AND PHRASES. C* algebra, quasiilpote op, Ritz n.

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