Character Space and Gelfand type representation of locally C^{*}-algebra

TL;DR

This paper develops a Gelfand-type representation for commutative unital locally C*-algebras, linking them to continuous functions on their character space, and explores their spectral properties and functional calculus.

Contribution

It introduces a novel approach to define the character space of locally C*-algebras and establishes a Gelfand representation connecting these algebras with function spaces.

Findings

Character space defined via inductive limits of topological spaces

Gelfand representation established for commutative unital locally C*-algebras

Spectral mapping theorem proved within this framework

Abstract

In this article, we identify a suitable approach to define the character space of a commutative unital locally $C^{\ast}$-algebra via the notion of the inductive limit of topological spaces. Also, we discuss topological properties of the character space. We establish the Gelfand type representation between a commutative unital locally $C^{\ast}$-algebra and the space of all continuous functions defined on its character space. Equivalently, we prove that every commutative unital locally $C^{\ast}$-algebra is identified with the locally $C^{\ast}$-algebra of continuous functions on its character space through the coherent representation of projective limit of $C^{\ast}$-algebras. Finally, we construct a unital locally $C^{\ast}$-algebra generated by a given locally bounded normal operator and show that its character space is homeomorphic to the local spectrum. Further, we define the functional calculus and prove spectral mapping theorem in this framework.