Zemánek, Jaroslav

Institute of Mathematics AS CR, v. v. i.

Commentationes Mathematicae Universitatis CarolinaeJaroslav ZemánekConcerning spectral characterizations of the radical in Banach algebrasCommentationes Mathematicae Universitatis Carolinae, Vol. 17 (1976), No. 4, 689--691Persistent URL: http://dml.cz/dmlcz/105729Terms of use:© Charles University in Prague, Faculty of Mathematics and Physics, 1976Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document mustcontain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://project.dml.czCOMMENTATIOKES MATHEMATICiffi TOIVEBSITATIS CAROLINAE 17,4 (1976) CONCERNING SPECTRAL CHARACTERIZATIONS OP THE RADICAL IN BANACH ALGEBRAS Jaroslav 2.EMÍNEK, Praha Abstract: An element r of a Banach algebra A belongs to the radical of A if ant only if 1(1 + q)r 1̂  * 0 for all q quas i-nilpotent in A. Key words: Spectral radius, the radical of a Banach algebra. AMS: 46H05 Ref. 2.: 7.976.11 We consider an arbitrary Banach algebra A over the com-plex field. For x in A, let 6* (x) be the spectrum (taken in the unitization of A if A has no unit) and Ixlg the spectral radius of the element x. Denote by N the set of quas i-nilpotent elements in A, i.e. N -=«Cx€A: 1x1^ - 0$ , and ly rad A the (Jacobson) radical of A. It is well-known that N^rad A, but this inclusion can often be proper. A characterization of algebras in which N « rad A is given in til (the set N is to be invariant under sums or, which is equivalent, under products). Thus although the radical is not - in general - simply the set of all quasi-nilpotents, it can nevertheless be characterized in terms of the spec-tral radius. One such characterization [21 is based on the observa-- 689 -tion that 6*(a + r) * ^ (a) for all a€ Af re rad A. We have shown in £21 that "if, conversely, 6*(a + r) « & (a) for all ae A ant some re A, then it must be re rad A* In factt the following theorem has appeared first in 121 alt-hough it was implicitly contained already in [11* Theorem 1, Let A be a Banaeh algebra * Suppose r€A is such that I a + r 1^ = 0 for all a€ N* Then re rad A« Another criterion has been known from early times of Banaeh algebras: if r€ A is such that I xr l# * 0 for all xfiA, then r€ rad A« How, Theorem 1 suggests that it should be possible to .restrict the range of x's in this multipli-cative criterion to some smaller subset of A being in some relation to the set N* We have remarked in [21 that it is not sufficients for trivial reasons, to require the condi-tion simply for all x€ N* However, it turns out that the appropriate restriction is to the elements of the form x =- 1 + a with a€ H. Indeed, the following result is a con-sequence of Theorem 1* Theorem 2, Let A be m Banaeh algebra* Suppose re A is such that I (1 * a)r 1$ - 0 for all aeK. Then r<s rad A. Proof* We show that I at + r tg = 0 for all ac N; then the conclusion will follow hy Ikeorem 1* Hence take an a€ H* It is enough to prove that, say, -1 does not belong to 6* (a + r)« But we have the decomposition 1 + a + r =- Q + a H l + E l - (1 + al^alr } where the element C 1 - (I + affair is quaei-nilpotent by assumption* It follows that the ele-- 690 -ment 1 + a + rf being represented as a product of two inver-tible elements, is invertibla as well. This completes the proof. We obtain similar corollaries as in f 23. Let us men-tion two of them. Corollary 1. If K is a Banaeh space operator such that I (1 * Q)H 1̂  -» 0 for all Q quasi-nilpotent, then R a 0. Corollary 2. The closed operator algebra generated by all the quasi-nilpotent operators on a Banaeh spaee is semi-simple. . R e f e r e n c e s 111 Z. SIJODKOWSKI, W. WOJTXffSKI, J . Z B l t o C . A note on qua&ini lpotent elements of a Banaeh a lgeb ra , B u l l . Acad. Poion* Sci .*Se>. Sci» Math. A s t r o -noiw B a y s , , t o appear . [ 2 1 J . ZEMiMEKi A note on the r a d i c a l of a Banaeh algebra f t o appear . Hatematick^ \ ls tav CSA? 2itna. 25 , 11567 Praha 1 Ceslcoslovensko (Oblatum 3 . 8 . 1976) 691 

English

Concerning spectral characterizations of the radical in Banach algebras

Czech digital mathematics library

http://dml.cz/bitstream/handle/10338.dmlcz/105729/CommentatMathUnivCarol_017-1976-4_7.pdf

Concerning spectral characterizations of the radical in Banach algebras

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.

Czech digital mathematics library