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The range of a ring homomorphism
from a commutative $C^{*}$-algebra

Author: Lajos Molnár
Journal: Proc. Amer. Math. Soc. 124 (1996), 1789-1794
MSC (1991): Primary 46J05, 46E25
MathSciNet review: 1307551
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Abstract: We prove that if a commutative semi-simple Banach algebra $\mathcal A $ is the range of a ring homomorphism from a commutative $C^{*}$-algebra, then $\mathcal A $ is $C^{*}$-equivalent, i.e. there are a commutative $C^{*}$-algebra $\mathcal B $ and a bicontinuous algebra isomorphism between $\mathcal A $ and $\mathcal B $. In particular, it is shown that the group algebras $L^{1}(\mathbb {R})$, $L^{1}(\mathbb {T})$ and the disc algebra $A(\mathbb {D})$ are not ring homomorphic images of $C^{*}$-algebras.

References [Enhancements On Off] (What's this?)

  • [Bur] R. B. Burckel, Characterizations of 𝐶(𝑋) among its subalgebras, Marcel Dekker, Inc., New York, 1972. Lecture Notes in Pure and Applied Mathematics, Vol. 6. MR 0442687
  • [Cro] R. W. Cross, On the continuous linear image of a Banach space, J. Austral. Math. Soc. Ser. A 29 (1980), no. 2, 219–234. MR 566287
  • [Cun] Joachim Cuntz, Locally 𝐶*-equivalent algebras, J. Functional Analysis 23 (1976), no. 2, 95–106. MR 0448088,
  • [FD] J. M. G. Fell and R. S. Doran, Representations of *-algebras, locally compact groups, and Banach *-algebraic bundles. Vol. 1, Pure and Applied Mathematics, vol. 125, Academic Press, Inc., Boston, MA, 1988. Basic representation theory of groups and algebras. MR 936628
  • [FW] P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441,
  • [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
  • [Mol] L. Molnár, Algebraic difference between $p$-classes of an $H^{*}$-algebra, Proc. Amer. Math. Soc. (to appear). CMP 94:17
  • [Pal] 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
  • [Rud] W. Rudin, Real and Complex Analysis, Tata McGraw-Hill Publishing Co. Ltd., New Delhi, 1983.

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Additional Information

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

Keywords: Ring homomorphism, commutative Banach algebra, Gelfand representation
Received by editor(s): November 21, 1994
Additional Notes: Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society