On the maximal ideal space of even quasicontinuous functions on the unit circle

TL;DR

This paper investigates the structure of the maximal ideal space of even quasicontinuous functions on the unit circle, revealing its detailed partitioning and fiber structure over related algebras.

Contribution

It provides a detailed analysis and description of the maximal ideal space of the algebra of even quasicontinuous functions, including fiber structure over related algebras.

Findings

Describes the maximal ideal space $M( ilde{QC})$ and its partitioning.

Analyzes fibers of $M(PQC)$ and $M(QC)$ over $M( ilde{QC})$.

Provides a detailed topological and algebraic structure of these spaces.

Abstract

Let $PQC$ stand for the set of all piecewise quasicontionus function on the unit circle, i.e., the smallest closed subalgebra of $L^\infty(\mathbb{T})$ which contains the classes of all piecewise continuous function $PC$ and all quasicontinuous functions $QC=(C+H^\infty)\cap(C+\overline{H^\infty})$. We analyze the fibers of the maximal ideal spaces $M(PQC)$ and $M(QC)$ over maximal ideals from $M(\widetilde{QC})$, where $\widetilde{QC}$ stands for the $C^*$-algebra of all even quasicontinous functions. The maximal ideal space $M(\widetilde{QC})$ is decribed and partitioned into various subsets corresponding to different descriptions of the fibers.