Let be a -Banach algebra with one element and a linear functional. Then the following statements are equivalent:
, and is multiplicative, that is, for everyone .
, and consists only of non-invertible elements.
for all , that is, for each lies in the spectrum of
Remarks
The conclusions are very simple. The non-trivial statement of the sentence is in the end .
For real Banach algebras the theorem is wrong. If the Banach algebra of continuous functions is defined by , then is a continuous linear functional. According to the mean value theorem of integral calculus , for each there is a with , and lies in the spectrum of , because has a zero, namely , and is therefore not invertible. Therefore fulfills the third point of the above sentence, but not the first, because integrating is, as is well known, not multiplicative.
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FF Bonsall, J. Duncan: Complete Normed Algebras . Springer-Verlag 1973, ISBN 3-5400-6386-2 .
Andrew M. Gleason: A characterization of maximal ideals . Journal d'Analyse Mathématique, Volume 19 (1967), pages 171-172.
Jean-Pierre Kahane, Wiesław Żelazko: A characterization of maximal ideals in commutative Banach algebras . Studia Mathematica , Volume 29 (1968), pages 339-343.