Best approximations, distance formulas and orthogonality in C*-algebras

TL;DR

This paper characterizes best approximations and distance formulas in unital C*-algebras, and explores orthogonality in Hilbert C*-modules, advancing understanding of approximation theory in operator algebras.

Contribution

It provides new characterizations for best approximations, distance formulas, and orthogonality in C*-algebras and Hilbert C*-modules.

Findings

Characterization of best approximation in unital C*-algebras

Formula for the distance from an element to a subspace

Characterization of Birkhoff-James orthogonality in Hilbert C*-modules

Abstract

For a unital $C^*$-algebra $\mathcal A$ and a subspace $\mathcal B$ of $\mathcal A$, a characterization for a best approximation to an element of $\mathcal A$ in $\mathcal B$ is obtained. As an application, a formula for the distance of an element of $\mathcal A$ from $\mathcal B$ has been obtained, when a best approximation of that element to $\mathcal B$ exists. Further, a characterization for Birkhoff-James orthogonality of an element of a Hilbert $C^*$-module to a subspace is obtained.