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A multiplicative Kowalski–Słodkowski theorem for
$C^\star $-algebras
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 66 / Issue 3 / September 2023
- Published online by Cambridge University Press:
- 02 November 2022, pp. 951-958
- Print publication:
- September 2023
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- Article
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We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a
$C^\star $-algebra, and if 
$\phi :A\to \mathbb C $ is a continuous function satisfying 
$ \phi (x)\phi (y) \in \sigma (xy) $ for all 
$x,y\in A$ (where 
$\sigma $ denotes the spectrum), then either 
$\phi $ is a character of A or 
$-\phi $ is a character of A.
MULTIPLICATIVE SPECTRAL FUNCTIONALS ON
$C(X)$
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 102 / Issue 2 / October 2020
- Published online by Cambridge University Press:
- 08 January 2020, pp. 303-307
- Print publication:
- October 2020
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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 
$C(X)$.
