Let C(Ω) be the algebra of all complex-valued continuous functions on a topological space Ω where C(Ω) contains unbounded functions. First it is shown that C(Ω) cannot have a Banach algebra norm. Then it is shown that, for certain Ω, C(Ω) cannot possess an (incomplete) normed algebra norm. In particular, this is so for where ℝ is the reals.
@article{bwmeta1.element.bwnjournal-article-smv111i1p97bwm,
author = {Bertram Yood},
title = {On the non-existence of norms for some algebras of functions},
journal = {Studia Mathematica},
volume = {108},
year = {1994},
pages = {97-101},
zbl = {0803.46027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv111i1p97bwm}
}
Yood, Bertram. On the non-existence of norms for some algebras of functions. Studia Mathematica, Tome 108 (1994) pp. 97-101. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv111i1p97bwm/
[00000] [1] W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. | Zbl 0093.12503
[00001] [2] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. | Zbl 0271.46039
[00002] [3] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960. | Zbl 0093.30001
[00003] [4] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.
[00004] [5] I. Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399-418. | Zbl 0033.18701
[00005] [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. | Zbl 0095.09702