We introduce and study an extension of the Roberts orthogonality, in the setting of C∗-algebras. More precisely, in a C∗-algebra A, for a,b∈A and a nonempty subset of A, say B, a is called B-Roberts orthogonal to b, denoted by a⊥BRb, if ∥a+bc∥=∥a−bc∥ for all c∈B. For certain special C∗-algebras, including the C∗-algebra of all 2×2 complex matrices, we obtain a nontrivial subset B such that the B-Roberts orthogonality coincides with the usual orthogonality. We also introduce a new concept of smoothness in normed linear spaces in terms of the additivity property of the usual Roberts orthogonality. A complete characterization of the same is obtained in the case of the classical ℓn∞ spaces.

Keywords