Gateaux derivative of C* norm

TL;DR

This paper derives the Gateaux derivative of the $C^*$-algebra norm, providing new insights and generalizations for subdifferential sets, smooth points, and orthogonality in operator and function spaces.

Contribution

It offers an explicit expression for the Gateaux derivative of the $C^*$-norm and extends known results on subdifferential sets and orthogonality in operator and function spaces.

Findings

Expression for Gateaux derivative of $C^*$-norm

Generalizations of subdifferential set results

Characterization of orthogonality conditions

Abstract

We find an expression for Gateaux derivative of the $C^*$-algebra norm. This gives us alternative proofs or generalizations of various known results on the closely related notions of subdifferential sets, smooth points and Birkhoff-James orthogonality for spaces $\mathscr B(\mathcal H)$ and $C_b(\Omega)$. We also obtain an expression for subdifferential sets of the norm function at $A\in\mathscr B(\mathcal H)$ and a characterization of orthogonality of an operator $A\in\mathscr B(\mathcal H, \mathcal K)$ to a subspace, under the condition $dist(A, \mathscr K(\mathcal H))< \|A\|$ and $dist(A, \mathscr K(\mathcal H, \mathcal K))< \|A\|$ respectively.