Li et al. [‘Weak 2-local isometries on uniform algebras and Lipschitz algebras’, Publ. Mat.63 (2019), 241–264] generalized the Kowalski–Słodkowski theorem by establishing the following spherical variant: let A be a unital complex Banach algebra and let \Delta : A \to \mathbb {C}$\Delta : A \to \mathbb {C}$ be a mapping satisfying the following properties:

1. (a) \Delta $\Delta$ is 1-homogeneous (that is, \Delta (\lambda x)=\lambda \Delta (x)$\Delta (\lambda x)=\lambda \Delta (x)$ for all x \in A$x \in A$ , \lambda \in \mathbb C$\lambda \in \mathbb C$ );

2. (b) \Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$\Delta (x)-\Delta (y) \in \mathbb {T}\sigma (x-y), \quad x,y \in A$ .

\Delta $\Delta$

\lambda _{0} \in \mathbb {T}$\lambda _{0} \in \mathbb {T}$

\lambda _{0}\Delta $\lambda _{0}\Delta$

\Delta (0)=0$\Delta (0)=0$

\Delta $\Delta$

\overline {\Delta (\mathbf {1})}\Delta $\overline {\Delta (\mathbf {1})}\Delta$

Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.