# The Banach algebra $U(X)$ on a zero-dimensional space

Document Type : Original Paper

Author

Department of mathematics, Faculty of Basic Sciences, Yasouj university, Yasouj, Iran

Abstract

In this research, for a zero-dimensional space $X$, a Banach subalgebra $U(X)$ of $C^{*}(X,\mathbb{C})$ is introduced. It is shown that $U(X)$ is the uniform closure of the subalgebras $C^{F}(X,\mathbb{C})$ and $C^{*}_{c}(X,\mathbb{C})$ of the Banach algebra $C^{*}(X,\mathbb{C})$. Moreover a necessary and sufficient condition for the coincidence of $U(X)$ and $C^{*}(X,\mathbb{C})$ is given. It is shown that $U(X)$ consists exactly of all $f\in C^{*}(X,\mathbb{C})$ each of which has an extension to

$\beta_{\circ}X$. Using this fact, an isometric isomorphism from $U(X)$ onto $C(\beta_{\circ}X,\mathbb{C})$ is defined. Finally, a description of the elements of $U(X)$ in terms of the inverse image of the closed subsets of $\mathbb{C}$ is given.

Keywords

Main Subjects

#### References

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### History

• Receive Date: 29 September 2022
• Revise Date: 17 December 2022
• Accept Date: 23 December 2022
• First Publish Date: 23 December 2022