## Banach Journal of Mathematical Analysis

### Vector-valued characters on vector-valued function algebras

Mortaza Abtahi

#### Abstract

Let $A$ be a commutative unital Banach algebra and let $X$ be a compact space. We study the class of $A$-valued function algebras on $X$ as subalgebras of $C(X,A)$ with certain properties. We introduce the notion of $A$-characters of an $A$-valued function algebra $\mathscr{A}$ as homomorphisms from $\mathscr{A}$ into $A$ that basically have the same properties as evaluation homomorphisms $\mathcal{E}_{x}:f\mapsto f(x)$, with $x\in X$. We show that, under certain conditions, there is a one-to-one correspondence between the set of $A$-characters of $\mathscr{A}$ and the set of characters of the function algebra $\mathfrak{A}=\mathscr{A}\cap C(X)$ of all scalar-valued functions in $\mathscr{A}$. For the so-called natural $A$-valued function algebras, such as $C(X,A)$ and $\operatorname{Lip}(X,A)$, we show that $\mathcal{E}_{x}$ ($x\in X$) are the only $A$-characters. Vector-valued characters are utilized to identify vector-valued spectra.

#### Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 608-620.

Dates
Accepted: 27 November 2015
First available in Project Euclid: 22 July 2016

https://projecteuclid.org/euclid.bjma/1469199412

Digital Object Identifier
doi:10.1215/17358787-3607486

Mathematical Reviews number (MathSciNet)
MR3528350

Zentralblatt MATH identifier
1365.46048

#### Citation

Abtahi, Mortaza. Vector-valued characters on vector-valued function algebras. Banach J. Math. Anal. 10 (2016), no. 3, 608--620. doi:10.1215/17358787-3607486. https://projecteuclid.org/euclid.bjma/1469199412

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