Gleason-Kahane-Żelazko theorem

The set of Gleason Kahane Żelazko , named after Andrew Gleason , Jean-Pierre Kahane and Wiesław Żelazko is a phrase from the mathematical branch of functional analysis . The theorem characterizes the multiplicative linear functionals on a - Banach algebra . ${\displaystyle \mathbb {C} }$

Formulation of the sentence

Let be a -Banach algebra with one element and a linear functional. Then the following statements are equivalent: ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle e}$ ${\displaystyle \varphi :A\rightarrow \mathbb {C} }$

1. ${\displaystyle \varphi \not =0}$, and is multiplicative, that is, for everyone . ${\displaystyle \varphi }$ ${\displaystyle \varphi (ab)=\varphi (a)\varphi (b)}$ ${\displaystyle a,b\in A}$
2. ${\displaystyle \varphi (e)=1}$, and consists only of non-invertible elements. ${\displaystyle {\rm {ker(\varphi )}}}$
3. ${\displaystyle \varphi (a)\in \sigma (a)}$for all , that is, for each lies in the spectrum of ${\displaystyle a\in A}$ ${\displaystyle a\in A}$ ${\displaystyle \varphi (a)}$ ${\displaystyle a}$

Remarks

• The conclusions are very simple. The non-trivial statement of the sentence is in the end . ${\displaystyle (1)\Rightarrow (2)\Rightarrow (3)}$ ${\displaystyle (3)\Rightarrow (1)}$
• 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. ${\displaystyle C_{\mathbb {R} }([0,1])}$ ${\displaystyle [0,1]\rightarrow \mathbb {R} }$ ${\displaystyle \varphi :C_{\mathbb {R} }([0,1])\rightarrow \mathbb {R} }$ ${\displaystyle \varphi (f):=\int _{0}^{1}f(t)\,{\rm {d}}t}$ ${\displaystyle \varphi }$ ${\displaystyle f\in C_{\mathbb {R} }([0,1])}$ ${\displaystyle t_{0}\in [0,1]}$ ${\displaystyle \varphi (f)=\int _{0}^{1}f(t)\,{\rm {d}}t=f(t_{0})}$ ${\displaystyle f(t_{0})}$ ${\displaystyle f}$ ${\displaystyle f-f(t_{0})\cdot 1}$ ${\displaystyle t_{0}}$ ${\displaystyle \varphi }$

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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.