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

Abstract

For a unital $C^{\ast }$-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 $C^{\ast }$-module to a subspace is obtained.

To appear in Journal of the Ramanujan Mathematical Society