## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 11, Number 1 (2017), 207-222.

### Order structure, multipliers, and Gelfand representation of vector-valued function algebras

Jorma Arhippainen, Jukka Kauppi, and Jussi Mattas

#### Abstract

We continue the study begun by the third author of ${C}^{\ast}$-Segal algebra-valued function algebras with an emphasis on the order structure. Our main result is a characterization theorem for ${C}^{\ast}$-Segal algebra-valued function algebras with an order unitization. As an intermediate step, we establish a function algebraic description of the multiplier module of arbitrary Segal algebra-valued function algebras. We also consider the Gelfand representation of these algebras in the commutative case.

#### Article information

**Source**

Banach J. Math. Anal., Volume 11, Number 1 (2017), 207-222.

**Dates**

Received: 2 November 2015

Accepted: 17 March 2016

First available in Project Euclid: 9 December 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1481274115

**Digital Object Identifier**

doi:10.1215/17358787-3784682

**Mathematical Reviews number (MathSciNet)**

MR3582396

**Zentralblatt MATH identifier**

1362.46051

**Subjects**

Primary: 46H05: General theory of topological algebras

Secondary: 46L05: General theory of $C^*$-algebras 46H10: Ideals and subalgebras

**Keywords**

vector-valued function algebra $C^{*}$-Segal algebra multiplier module order unitization Gelfand representation

#### Citation

Arhippainen, Jorma; Kauppi, Jukka; Mattas, Jussi. Order structure, multipliers, and Gelfand representation of vector-valued function algebras. Banach J. Math. Anal. 11 (2017), no. 1, 207--222. doi:10.1215/17358787-3784682. https://projecteuclid.org/euclid.bjma/1481274115