Bulletin of the Belgian Mathematical Society - Simon Stevin

An approach to Gelfand theory for arbitrary Banach algebras

G.A. Bagheri-Bardi and F. Behrouzi

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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

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

First available in Project Euclid: 9 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A15: Invariant subspaces [See also 47A46]
Secondary: 46A32: Spaces of linear operators; topological tensor products; approximation properties [See also 46B28, 46M05, 47L05, 47L20] 47D20

Banach algebra $C^*$-algebra Gelfand theory


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

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