The strong radical and finite-dimensional ideals

Author:
Bertram Yood

Journal:
Proc. Amer. Math. Soc. **130** (2002), 139-143

MSC (2000):
Primary 46H10; Secondary 16D25

Published electronically:
May 23, 2001

MathSciNet review:
1855630

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a semi-prime Banach algebra with strong radical (intersection of its two-sided modular maximal ideals). A minimal left or right ideal of is infinite-dimensional if and only if . Thus all minimal one-sided ideals in are finite-dimensional if is strongly semi-simple.

**1.**Emil Artin, Cecil J. Nesbitt, and Robert M. Thrall,*Rings with Minimum Condition*, University of Michigan Publications in Mathematics, no. 1, University of Michigan Press, Ann Arbor, Mich., 1944. MR**0010543****2.**Frank F. Bonsall and John Duncan,*Complete normed algebras*, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80. MR**0423029****3.**J. W. Calkin,*Two-sided ideals and congruences in the ring of bounded operators in Hilbert space*, Ann. of Math. (2)**42**(1941), 839–873. MR**0005790****4.**Leoni Dalla, S. Giotopoulos, and Nelli Katseli,*The socle and finite-dimensionality of a semiprime Banach algebra*, Studia Math.**92**(1989), no. 2, 201–204. MR**986948****5.**I. N. Herstein,*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205****6.**T. W. Palmer,*Classes of nonabelian, noncompact, locally compact groups*, Rocky Mountain J. Math.**8**(1978), no. 4, 683–741. MR**513952**, 10.1216/RMJ-1978-8-4-683**7.**Theodore W. Palmer,*Banach algebras and the general theory of *-algebras. Vol. I*, Encyclopedia of Mathematics and its Applications, vol. 49, Cambridge University Press, Cambridge, 1994. Algebras and Banach algebras. MR**1270014****8.**C. E. Rickart,*General theory of Banach algebras*, Van Nostrand, Princeton, 1960. MR 22:5903**9.**I. E. Segal,*The group algebra of a locally compact group*, Trans. Amer. Math. Soc.**61**(1947), 69–105. MR**0019617**, 10.1090/S0002-9947-1947-0019617-4**10.**M. R. F. Smythe,*On problems of Olubummo and Alexander*, Proc. Royal Irish Acad. 80A (1980), 69-74.**11.**Bertram Yood,*Ideals in topological rings*, Canad. J. Math.**16**(1964), 28–45. MR**0158279****12.**Bertram Yood,*Closed prime ideals in topological rings*, Proc. London Math. Soc. (3)**24**(1972), 307–323. MR**0298423****13.**Bertram Yood,*Finite-dimensional ideals in Banach algebras*, Colloq. Math.**63**(1992), no. 2, 295–301. MR**1180641**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
46H10,
16D25

Retrieve articles in all journals with MSC (2000): 46H10, 16D25

Additional Information

**Bertram Yood**

Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

DOI:
https://doi.org/10.1090/S0002-9939-01-06049-X

Received by editor(s):
January 7, 2000

Received by editor(s) in revised form:
June 10, 2000

Published electronically:
May 23, 2001

Communicated by:
Dale Alspach

Article copyright:
© Copyright 2001
American Mathematical Society