Automatic surjectivity of ring homomorphisms on -algebras and algebraic differences among some group algebras of compact groups

Author:
Lajos Molnár

Journal:
Proc. Amer. Math. Soc. **128** (2000), 125-134

MSC (1991):
Primary 46K15, 47D50, 47B49; Secondary 43A15, 43A22

Published electronically:
June 30, 1999

MathSciNet review:
1616645

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we present two automatic surjectivity results concerning ring homomorphisms between -classes of an -algebra which, in some sense, improve the main theorem in a recent paper by the author (*Proc. Amer. Math. Soc.* **124** (1996), 169-175) quite significantly. Furthermore, we apply our results to show that for arbitrary infinite compact groups , no quotient ring of is isomorphic to , a statement we conjecture to be true for every pair of group rings corresponding to different exponents .

**[AD]**J. Aczél and J. Dhombres,*Functional equations in several variables*, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. With applications to mathematics, information theory and to the natural and social sciences. MR**1004465****[Amb]**Warren Ambrose,*Structure theorems for a special class of Banach algebras*, Trans. Amer. Math. Soc.**57**(1945), 364–386. MR**0013235**, 10.1090/S0002-9947-1945-0013235-8**[Arn]**B. H. Arnold,*Rings of operators on vector spaces*, Ann. of Math. (2)**45**(1944), 24–49. MR**0009425****[Dra]**D.D. Draghia,*Continuitate in Algebre Banach*, Editure didactica si pedagogica, Bucuresti, 1995.**[Eid]**M. Eidelheit,*On isomorphisms of rings of linear operators*, Studia Math.**9**(1940), 97–105 (English, with Ukrainian summary). MR**0004725****[GK]**I. C. Gohberg and M. G. Kreĭn,*Introduction to the theory of linear nonselfadjoint operators*, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. MR**0246142****[HR]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups*, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR**0262773****[Kap]**Irving Kaplansky,*Ring isomorphisms of Banach algebras*, Canadian J. Math.**6**(1954), 374–381. MR**0062960****[Kuc]**Marek Kuczma,*An introduction to the theory of functional equations and inequalities*, Prace Naukowe Uniwersytetu Śląskiego w Katowicach [Scientific Publications of the University of Silesia], vol. 489, Uniwersytet Śląski, Katowice; Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1985. Cauchy’s equation and Jensen’s inequality; With a Polish summary. MR**788497****[Mol1]**Lajos Molnár,*𝑝-classes of an 𝐻*-algebra and their representations*, Acta Sci. Math. (Szeged)**58**(1993), no. 1-4, 411–423. MR**1264246****[Mol2]**Lajos Molnár,*Algebraic difference between 𝑝-classes of an 𝐻*-algebra*, Proc. Amer. Math. Soc.**124**(1996), no. 1, 169–175. MR**1291787**, 10.1090/S0002-9939-96-03048-1**[Mol3]**Lajos Molnár,*The range of a ring homomorphism from a commutative 𝐶*-algebra*, Proc. Amer. Math. Soc.**124**(1996), no. 6, 1789–1794. MR**1307551**, 10.1090/S0002-9939-96-03236-4**[New]**J. D. Newburgh,*The variation of spectra*, Duke Math. J.**18**(1951), 165–176. MR**0051441****[SF]**Parfeny P. Saworotnow and John C. Friedell,*Trace-class for an arbitrary 𝐻*-algebra*, Proc. Amer. Math. Soc.**26**(1970), 95–100. MR**0267402**, 10.1090/S0002-9939-1970-0267402-9**[SG]**Parfeny P. Saworotnow and George R. Giellis,*Continuity and linearity of centralizers on a complemented algebra.*, Proc. Amer. Math. Soc.**31**(1972), 142–146. MR**0288585**, 10.1090/S0002-9939-1972-0288585-2**[Sem]**Peter Šemrl,*Isomorphisms of standard operator algebras*, Proc. Amer. Math. Soc.**123**(1995), no. 6, 1851–1855. MR**1242104**, 10.1090/S0002-9939-1995-1242104-8

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
46K15,
47D50,
47B49,
43A15,
43A22

Retrieve articles in all journals with MSC (1991): 46K15, 47D50, 47B49, 43A15, 43A22

Additional Information

**Lajos Molnár**

Affiliation:
Institute of Mathematics, Lajos Kossuth University, 4010 Debrecen, P. O. Box 12, Hungary

Email:
molnarl@math.klte.hu

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-04974-6

Keywords:
Ring homomorphism,
$H^{*}$-algebra,
$p$-class,
compact group,
group algebra,
automatic surjectivity

Received by editor(s):
March 4, 1997

Received by editor(s) in revised form:
March 10, 1998

Published electronically:
June 30, 1999

Additional Notes:
This paper was completed when the author, holding a Hungarian State Eötvös Scholarship, was a visitor at the University of Maribor, Slovenia. This research was supported also by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T–016846 F–019322.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1999
American Mathematical Society