Complexifications of real Banach spaces and their isometries.

Every norm ‖ ⋅ ‖ on a real Banach space X induces a minimal norm on the complex linear space C X = X + i X = { x + i y : x , y ∈ X } by ‖ x + i y ‖ C = sup ⁡ { ‖ x cos ⁡ θ + y sin ⁡ θ ‖ : θ ∈ [ 0 , 2 π ] }. In this note we show that if X is finite-dimensional there is a decomposition X = X 1 ⊕ ⋯ ⊕ X k into subspaces such that the isometry group of ‖ ⋅ ‖ C is generated by that of ‖ ⋅ ‖ and operators of the form e i θ 1 I n 1 ⊕ ⋯ ⊕ e i θ k I n k acting on C X = C X 1 ⊕ ⋯ ⊕ C X k. Various applications are given, in particular to isometries of numerical radius. [ABSTRACT FROM AUTHOR]