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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 cheickkader89@hotmail.com
R. BRITS*
Affiliation:
Department of Mathematics,University of Johannesburg, South Africa email rbrits@uj.ac.za
*
✉rbrits@uj.ac.za
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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., ‘Multiplicative maps into the spectrum’, Studia Math. 239 (2017), 55–66.CrossRefGoogle Scholar
Touré, C., Schulz, F. and Brits, R., ‘Some character generating functions on Banach algebras’, J. Math. Anal. Appl. 468 (2018), 704–715.CrossRefGoogle Scholar