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MULTIPLICATIVE SPECTRAL FUNCTIONALS ON $C(X)$

Part of: Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  08 January 2020

C. TOURÉ  and
R. BRITS
C. TOURÉ
Affiliation:
Department of Mathematics,University of Johannesburg, South Africa email [email protected]
R. BRITS*
Affiliation:
Department of Mathematics,University of Johannesburg, South Africa email [email protected]
*
✉[email protected]
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Abstract

If $A$ is a commutative $C^{\star }$-algebra and if $\unicode[STIX]{x1D719}:A\rightarrow \mathbb{C}$ is a continuous multiplicative functional such that $\unicode[STIX]{x1D719}(x)$ belongs to the spectrum of $x$ for each $x\in A$, then $\unicode[STIX]{x1D719}$ is linear and hence a character of $A$. This establishes a multiplicative Gleason–Kahane–Żelazko theorem for $C(X)$.

Keywords

spectrummultiplicative functionallinear functional

MSC classification

Primary: 46J10: Banach algebras of continuous functions, function algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 303 - 307
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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References

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Touré, C., Schulz, F. and Brits, R., ‘Some character generating functions on Banach algebras’, J. Math. Anal. Appl. 468 (2018), 704–715.CrossRefGoogle Scholar