Gateaux derivative of C* norm

Abstract

We find an expression for Gateaux derivative of the $C^{\ast }$-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 $\mathcal{B}(\mathcal{H})$ and $C_b(\Omega )$. We also obtain an expression for subdifferential sets of the norm function at $A\in \mathcal{B}(\mathcal{H})$ and a characterization of orthogonality of an operator $A\in \mathcal{B}(\mathcal{H},\mathcal{K})$ to a subspace, under the condition $dist(A,\mathcal{K}(\mathcal{H}))<\Vert A\Vert$ and $dist(A,\mathcal{K}(\mathcal{H},\mathcal{K}))<\Vert A\Vert$ respectively.