Banach Journal of Mathematical Analysis

Vector-valued characters on vector-valued function algebras

Mortaza Abtahi

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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 A as homomorphisms from A into A that basically have the same properties as evaluation homomorphisms Ex:ff(x), with xX. We show that, under certain conditions, there is a one-to-one correspondence between the set of A-characters of A and the set of characters of the function algebra A=AC(X) of all scalar-valued functions in A. For the so-called natural A-valued function algebras, such as C(X,A) and Lip(X,A), we show that Ex (xX) are the only A-characters. Vector-valued characters are utilized to identify vector-valued spectra.

Article information

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

Received: 23 May 2015
Accepted: 27 November 2015
First available in Project Euclid: 22 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25]
Secondary: 46J20: Ideals, maximal ideals, boundaries 46H10: Ideals and subalgebras 46E40: Spaces of vector- and operator-valued functions

Algebras of continuous vector-valued functions Banach function algebras vector-valued function algebras characters maximal ideals


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.

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