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

Bertram Yood

ArticleOriginal scientific text

Title

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

Authors ¹

Affiliations

Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802, U.S.A.

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.

W. G. Bade and P. C. Curtis Jr., Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608.
F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, 1973.
L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand, Princeton, 1960.
E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer, New York, 1965.
I. Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399-418.
C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960.

Title

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

Affiliations

Abstract

Bibliography