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 c0- andℓ1-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.