## Bulletin of the Belgian Mathematical Society - Simon Stevin

### An approach to Gelfand theory for arbitrary Banach algebras

#### Abstract

Let $A$ be a Banach algebra. We say that a pair $(\mathcal{G},\mathcal{U})$ is a (topologically Gelfand theory) Gelfand theory for $A$ if the following hold: (G1) $\mathcal{U}$ is a C*-algebra and $\mathcal{G}:A\to \mathcal{U}$ is a homomorphism which induces the (homeomorphism) bijection $\pi\mapsto \pi\circ \mathcal{G}$ from $\widehat{\mathcal{U}}$ onto $\widehat{\mathcal{A}}$; (G2) for every maximal modular left ideal $L$, $\mathcal{G}(A)\not\subseteq L$. We show that this definition is equivalent to the usual definition of gelfand theory in the commutative case. We prove that many properties of Gelfand theory of commutative Banach algebras remain true for Gelfand theories of arbitrary Banach algebras. We show that unital homogeneous Banach algebras and postliminal C*-algebras have unique Gelfand theories (up to an appropriate notion of uniqueness ).

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 16, Number 4 (2009), 637-646.

Dates
First available in Project Euclid: 9 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1257776239

Digital Object Identifier
doi:10.36045/bbms/1257776239

Mathematical Reviews number (MathSciNet)
MR2583551

Zentralblatt MATH identifier
1192.46044

#### Citation

Behrouzi, F.; Bagheri-Bardi, G.A. An approach to Gelfand theory for arbitrary Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 4, 637--646. doi:10.36045/bbms/1257776239. https://projecteuclid.org/euclid.bbms/1257776239