On the non-existence of norms for some algebras of functions

Bertram Yood

Studia Mathematica (1994)

  • Volume: 111, Issue: 1, page 97-101
  • ISSN: 0039-3223

Abstract

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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 Ω = ℝ n where ℝ is the reals.

How to cite

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Yood, Bertram. "On the non-existence of norms for some algebras of functions." Studia Mathematica 111.1 (1994): 97-101. <http://eudml.org/doc/216122>.

@article{Yood1994,
abstract = {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 $Ω = ℝ^n$ where ℝ is the reals.},
author = {Yood, Bertram},
journal = {Studia Mathematica},
keywords = {non-existence of norms; algebra of all complex-valued continuous functions on a topological space; unbounded functions},
language = {eng},
number = {1},
pages = {97-101},
title = {On the non-existence of norms for some algebras of functions},
url = {http://eudml.org/doc/216122},
volume = {111},
year = {1994},
}

TY - JOUR
AU - Yood, Bertram
TI - On the non-existence of norms for some algebras of functions
JO - Studia Mathematica
PY - 1994
VL - 111
IS - 1
SP - 97
EP - 101
AB - 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 $Ω = ℝ^n$ where ℝ is the reals.
LA - eng
KW - non-existence of norms; algebra of all complex-valued continuous functions on a topological space; unbounded functions
UR - http://eudml.org/doc/216122
ER -

References

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  1. [1] W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. Zbl0093.12503
  2. [2] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973. Zbl0271.46039
  3. [3] L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960. Zbl0093.30001
  4. [4] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965. 
  5. [5] I. Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399-418. Zbl0033.18701
  6. [6] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960. Zbl0095.09702

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