A note on the Mazur–Ulam property of almost-CL-spaces

We introduce the (T)-property, and prove that every Banach space with the (T)-property has the Mazur–Ulam property (for short, MUP). We obtain as its immediate applications that almost-CL-spaces admitting a smooth point (in particular, separable almost-CL-spaces) and a two-dimensional space whose unit sphere is a hexagon have the MUP. Furthermore, we discuss the stability of the spaces having the MUP derived from the c 0 - and ℓ 1 -sums, and show that the space C ( K , X ) of the vector-valued continuous functions has the MUP, where X is a separable almost-CL-space and K is a compact metric space.